Interesting integer sequences from the OEIS!

Introduction

Interesting integer sequences from the On-Line Encyclopedia of Integer Sequences (OEIS).

Notes

Radix-independent

These are interesting regardless of the base they're represented in.

Figurate

OEIS Description First terms
A000217 triangular ,0,1,3,6,10,15,21,28,36,45,55,66,78,91,105,120,136,153,171,190,210,231,253,
A000290 squares ,0,1,4,9,16,25,36,49,64,81,100,121,144,169,196,225,256,289,324,361,400,441,
A000326 pentagonal ,0,1,5,12,22,35,51,70,92,117,145,176,210,247,287,330,376,425,477,532,590,651,
A000384 hexagonal; every other triangular ,0,1,6,15,28,45,66,91,120,153,190,231,276,325,378,435,496,561,630,703,780,861,
A000566 heptagonal ,0,1,7,18,34,55,81,112,148,189,235,286,342,403,469,540,616,697,783,874,970,
A000567 octagonal ,0,1,8,21,40,65,96,133,176,225,280,341,408,481,560,645,736,833,936,
A001106 nonagonal ,0,1,9,24,46,75,111,154,204,261,325,396,474,559,651,750,856,969,
A001107 decagonal ,0,1,10,27,52,85,126,175,232,297,370,451,540,637,742,855,976,
A051624 dodecagonal ,0,1,12,33,64,105,156,217,288,369,460,561,672,793,924,
A005448 centered triangular ,1,4,10,19,31,46,64,85,109,136,166,199,235,274,316,361,409,460,514,571,631,
A001844 centered squares; sums of 2 consecutive squares ,1,5,13,25,41,61,85,113,145,181,221,265,313,365,421,481,545,613,685,761,841,
A005891 centered pentagonal ,1,6,16,31,51,76,106,141,181,226,276,331,391,456,526,601,681,766,856,951,
A003215 centered hexagonal; differences of 2 consecutive cubes ,1,7,19,37,61,91,127,169,217,271,331,397,469,547,631,721,817,919,
A069099 centered heptagonal ,1,8,22,43,71,106,148,197,253,316,386,463,547,638,736,841,953,
A016754 centered octagonal; odd squares ,1,9,25,49,81,121,169,225,289,361,441,529,625,729,841,961,
A060544 centered nonagonal; every third triangular ,1,10,28,55,91,136,190,253,325,406,496,595,703,820,946,
A062786 centered decagonal ,1,11,31,61,101,151,211,281,361,451,551,661,781,911,
A003154 centered dodecagonal; star numbers ,1,13,37,73,121,181,253,337,433,541,661,793,937,
A000292 tetrahedral; triangular pyramidal ,0,1,4,10,20,35,56,84,120,165,220,286,364,455,560,680,816,969,
A000578 cubes ,0,1,8,27,64,125,216,343,512,729,
A005900 octahedral ,0,1,6,19,44,85,146,231,344,489,670,891,
A006566 dodecahedral ,0,1,20,84,220,455,816,
A006564 icosahedral ,1,12,48,124,255,456,742,
A005894 centered tetrahedral ,1,5,15,35,69,121,195,295,425,589,791,
A005898 centered cubes; sums of 2 consecutive cubes ,1,9,35,91,189,341,559,855,
A001845 centered octahedral ,1,7,25,63,129,231,377,575,833,
A005904 centered dodecahedral ,1,33,155,427,909,
A005902 centered icosahedral ,1,13,55,147,309,561,923,
A000330 square pyramidal ,0,1,5,14,30,55,91,140,204,285,385,506,650,819,
A002411 pentagonal pyramidal ,0,1,6,18,40,75,126,196,288,405,550,726,936,
A002412 hexagonal pyramidal ,0,1,7,22,50,95,161,252,372,525,715,946,
A002413 heptagonal pyramidal ,0,1,8,26,60,115,196,308,456,645,880,
A002414 octagonal pyramidal ,1,9,30,70,135,231,364,540,765,
A007584 nonagonal pyramidal ,0,1,10,34,80,155,266,420,624,885,
A007585 decagonal pyramidal ,0,1,11,38,90,175,301,476,708,
A007588 stellated octahedron; stella octangula ,0,1,14,51,124,245,426,679,

Intersections of figurate numbers – polygonal and polygonal

OEIS Description First terms
A001110 triangular and square ,0,1,36,
A014979 triangular and pentagonal ,0,1,210,
A046194 triangular and heptagonal ,1,55,
A046183 triangular and octagonal ,1,21,
A048909 triangular and nonagonal ,1,325,
A133216 triangular and decagonal ,0,1,10,
A036353 square and pentagonal ,0,1,
A046177 square and hexagonal ,1,
A036354 square and heptagonal ,1,81,
A036428 square and octagonal ,1,225,
A036411 square and nonagonal ,1,9,
A046180 pentagonal and hexagonal ,1,
A048900 pentagonal and heptagonal ,1,
A046189 pentagonal and octagonal ,1,176,
A048915 pentagonal and nonagonal ,1,651,
A202563 pentagonal and decagonal ,1,
A048903 hexagonal and heptagonal ,1,
A046192 hexagonal and octagonal ,1,
A048918 hexagonal and nonagonal ,1,325,
A203134 hexagonal and decagonal ,1,
A048906 heptagonal and octagonal ,1,
A048921 heptagonal and nonagonal ,1,
A203408 heptagonal and decagonal ,1,540,
A048924 octagonal and nonagonal ,1,
A203624 octagonal and decagonal ,1,
A203627 nonagonal and decagonal ,1,

(The only “square and decagonal” numbers are 0 and 1.)

Intersections of figurate numbers – polygonal and centered polygonal

OEIS Description First terms
A128862 triangular and centered triangular ,1,10,136,
A254628 triangular and centered pentagonal ,1,6,276,
A253880 triangular and centered heptagonal ,1,253,
A280113 triangular and centered decagonal ,1,
A249483 square and centered triangular ,1,4,64,361,
A254333 square and centered pentagonal ,1,16,
A006051 square and centered hexagonal ,1,169,
A254711 pentagonal and centered square ,1,5,145,925,
A128917 pentagonal and centered pentagonal ,1,51,
A254138 pentagonal and centered hexagonal ,1,
A254654 pentagonal and centered heptagonal ,1,22,
A254285 hexagonal and centered triangular ,1,
A006244 hexagonal and centered hexagonal ,1,91,
A253716 hexagonal and centered heptagonal ,1,
A272399 hexagonal and centered nonagonal; every 6th triangular ,1,28,91,190,325,496,703,946,
A254676 heptagonal and centered triangular ,1,235,
A254230 heptagonal and centered square ,1,
A254966 heptagonal and centered hexagonal ,1,7,469,
A128919 heptagonal and centered heptagonal ,1,148,
A253823 octagonal and centered triangular ,1,
A254896 octagonal and centered square ,1,481,
A253923 octagonal and centered pentagonal ,1,
A254857 octagonal and centered heptagonal ,1,8,736,
A133142 decagonal and centered square ,1,

(triangular and centered hexagonal numbers = hexagonal and centered hexagonal numbers)

Intersections of figurate – centered polygonal and centered polygonal

OEIS Description First terms
A131750 centered triangular and centered square ,1,85,
A131751 centered triangular and centered pentagonal ,1,31,
A107118 centered triangular and centered hexagonal ,1,19,631,
A253689 centered triangular and centered heptagonal ,1,316,
A253675 centered triangular and centered octagonal ,1,361,
A107075 centered square and centered pentagonal ,1,181,
A253599 centered square and centered heptagonal ,1,841,
A133141 centered pentagonal and centered hexagonal ,1,361,
A253622 centered pentagonal and centered heptagonal ,1,106,
A253579 centered pentagonal and centered octagonal ,1,
A253546 centered hexagonal and centered heptagonal ,1,547,
A253514 centered heptagonal and centered octagonal ,1,841,

Intersections of figurate – other

OEIS Description First terms
A027568 triangular and tetrahedral ,0,1,10,120,
A039596 triangular and square pyramidal ,0,1,55,91,
A003556 square and tetrahedral ,0,1,4,
A344376 octagonal and octagonal pyramidal ,0,1,
A307491 centered triangular and tetrahedral ,1,4,10,
A307492 centered square and square pyramidal ,1,5,

Note: A156712 (“Star numbers (A003154) that are also triangular numbers (A000217)”) does not match A003154.

Primes – sums

OEIS Description First terms
A001043 sums of 2 consecutive primes ,5,8,12,18,24,30,36,42,52,60,68,78,84,90,100,112,120,128,138,144,152,162,172,
A034961 sums of 3 consecutive primes ,10,15,23,31,41,49,59,71,83,97,109,121,131,143,159,173,187,199,211,223,235,
A034963 sums of 4 consecutive primes ,17,26,36,48,60,72,88,102,120,138,152,168,184,202,220,240,258,272,290,306,324,
A034964 sums of 5 consecutive primes ,28,39,53,67,83,101,119,139,161,181,199,221,243,263,287,311,331,351,373,395,
A127333 sums of 6 consecutive primes ,41,56,72,90,112,132,156,180,204,228,252,280,304,330,358,384,410,434,462,492,
A127334 sums of 7 consecutive primes ,58,75,95,119,143,169,197,223,251,281,311,341,371,401,431,463,493,523,559,593,
A127335 sums of 8 consecutive primes ,77,98,124,150,180,210,240,270,304,340,372,408,442,474,510,546,582,620,660,
A127336 sums of 9 consecutive primes ,100,127,155,187,221,253,287,323,363,401,439,479,515,553,593,635,679,721,763,
A127337 sums of 10 consecutive primes ,129,158,192,228,264,300,340,382,424,468,510,552,594,636,682,732,780,824,870,
A127338 sums of 11 consecutive primes ,160,195,233,271,311,353,399,443,491,539,583,631,677,725,779,833,883,931,979,
A127339 sums of 12 consecutive primes ,197,236,276,318,364,412,460,510,562,612,662,714,766,822,880,936,990,
A054996 sums of 1+ consecutive primes in 1 way ,2,3,7,8,10,11,12,13,15,18,19,24,26,28,29,30,37,39,42,43,47,48,49,52,56,58,61,
A054997 sums of 1+ consecutive primes in 2 ways ,5,17,23,31,36,53,59,60,67,71,72,90,97,100,101,109,112,119,120,127,131,138,
A054998 sums of 1+ consecutive primes in 3 ways ,41,83,197,199,223,240,251,281,287,340,371,401,439,491,510,593,660,733,803,
A054999 sums of 1+ consecutive primes in 4 ways (none below 1000)
A055000 sums of 1+ consecutive primes in 5 ways ,311,863,
A055001 sums of 1+ consecutive primes in 6 ways (none below 1000)
A034707 sums of 1+ consecutive primes in 1+ ways ,2,3,5,7,8,10,11,12,13,15,17,18,19,23,24,26,28,29,30,31,36,37,39,41,42,43,47,
A309770 sums of 1+ consecutive primes in 2+ ways ,5,17,23,31,36,41,53,59,60,67,71,72,83,90,97,100,101,109,112,119,120,127,131,
A050936 sums of 2+ consecutive primes in 1+ ways ,5,8,10,12,15,17,18,23,24,26,28,30,31,36,39,41,42,48,49,52,53,56,58,59,60,67,
A067372 sums of 2+ consecutive primes in 2+ ways ,36,41,60,72,83,90,100,112,119,120,138,143,152,180,187,197,199,204,210,221,
A067373 sums of 2+ consecutive primes in 3+ ways ,240,287,311,340,371,510,660,803,863,864,931,961,990,
A067374 sums of 2+ consecutive primes in 4+ ways ,311,863,
A054845 number of ways representing n as the sum of 1+ consecutive primes 0,0,1,1,0,2,0,1,1,0,1,1,1,1,0,1,0,2,1,1,0,0,0,2,1,0,1,0,1,1,1,2,0,0,0,0,2,1,0,
A054859 smallest number expressible as the sum of 1+ consecutive primes in exactly n ways ,1,2,5,41,1151,311,34421,218918,3634531,48205429,1798467197,12941709050,
A007504 sums of first n primes ,0,2,5,10,17,28,41,58,77,100,129,160,197,238,281,328,381,440,501,568,639,712,
A014284 sums of first n noncomposite numbers ,1,3,6,11,18,29,42,59,78,101,130,161,198,239,282,329,382,441,502,569,640,713,
A062198 sums of first n semiprimes ,4,10,19,29,43,58,79,101,126,152,185,219,254,292,331,377,426,477,532,589,647,
A013916 sum of first a primes is prime ,1,2,4,6,12,14,60,64,96,100,102,108,114,122,124,130,132,146,152,158,162,178,
A092189 sum of first a semiprimes is a semiprime ,1,2,6,11,12,13,16,20,24,25,29,34,38,41,42,43,50,53,58,61,65,66,68,77,100,102,
A045345 a divides the sum of the first a primes ,1,23,53,853,
A179859 a divides the sum of the first a noncomposite numbers ,1,3,7,225,487,735,
A173663 a divides the sum of the first a semiprimes ,1,2,9,19,29,44,632,
A069484 sums of squares of 2 consecutive primes ,13,34,74,170,290,458,650,890,
A133529 sums of squares of 3 consecutive primes ,38,83,195,339,579,819,
A133524 sums of squares of 4 consecutive primes ,87,204,364,628,940,
A024450 sums of squares of first n primes ,4,13,38,87,208,377,666,
A098561 sum of squares of first a primes is prime ,2,18,26,36,68,78,144,158,164,174,192,212,216,236,264,288,294,338,344,356,384,
A111441 sum of squares of first a primes is a multiple of a ,1,19,37,455,509,575,
A098999 sums of cubes of first n primes ,8,35,160,503,
A098563 sum of cubes of first a primes is prime ,4,8,38,48,98,102,118,128,130,132,156,168,172,178,180,190,202,208,308,346,358,
A122140 sum of cubes of first a primes is a multiple of a ,1,25,537,661,
A054735 sums of twin prime pairs ,8,12,24,36,60,84,120,144,204,216,276,300,360,384,396,456,480,540,564,624,696,
A002375 number of unordered decompositions of 2n into sum of 2 odd primes ,0,0,1,1,2,1,2,2,2,2,3,3,3,2,3,2,4,4,2,3,4,3,4,5,4,3,5,3,4,6,3,5,6,2,5,6,5,5,
A001172 smallest number with exactly n unordered decompositions into sum of 2 odd primes ,0,6,10,22,34,48,60,78,84,90,114,144,120,168,180,234,246,288,240,210,324,300,
A082917 numbers with more unordered decompositions into 2 odd primes than any smaller number ,6,10,22,34,48,60,78,84,90,114,120,168,180,210,300,330,390,420,510,630,780,
A007506 primes p such that the sum of all primes ≤ p is a multiple of p ,2,5,71,
A013917 primes p such that the sum of all primes ≤ p is prime ,2,3,7,13,37,43,281,311,503,541,557,593,619,673,683,733,743,839,881,929,953,
A013918 primes that are the sum of the first k primes for any k ,2,5,17,41,197,281,
A051395 a2 is a sum of 4 consecutive primes ,6,18,24,42,48,70,144,252,258,358,378,388,396,428,486,506,510,558,608,644,864,
A071602 sum of the reverses of the first n primes ,2,5,10,17,28,59,130,221,253,345,358,431,445,479,553,588,683,699,775,792,829,
A254325 sequence of semiprimes with all cumulating sums being semiprime ,4,6,15,26,55,111,237,469,926,
A008472 sum of distinct prime factors of n ,0,2,3,2,5,5,7,2,3,7,11,5,13,9,8,2,17,5,19,7,10,13,23,5,5,15,3,9,29,10,31,2,
A001414 sum of prime factors of n (with multiplicity) ,0,2,3,4,5,5,7,6,6,7,11,7,13,9,8,8,17,8,19,9,10,13,23,9,10,15,9,11,29,10,31,
A006145 sum of distinct prime factors is the same for a and a+1; Ruth-Aaron numbers (1) ,5,24,49,77,104,153,369,492,714,
A039752 sum of prime factors (with multiplicity) is the same for a and a+1; Ruth-Aaron numbers (2) ,5,8,15,77,125,714,948,

Primes – other

OEIS Description First terms
A000040 primes ,2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,
A001358 semiprimes ,4,6,9,10,14,15,21,22,25,26,33,34,35,38,39,46,49,51,55,57,58,62,65,69,74,77,
A006562 balanced primes; average of previous and following ,5,53,157,173,211,257,263,373,563,593,607,653,733,947,977,
A088054 factorial primes; primes of form k!±1 ,2,3,5,7,23,719,
A055490 factorial primes; primes of form k!−1 ,5,23,719,
A088332 factorial primes; primes of form k!+1 ,2,3,7,
A088412 factorial prime, a!−1 or a!+1 is ,1,2,3,4,6,7,11,12,14,27,30,32,33,37,38,41,73,77,94,116,154,166,320,324,340,
A002982 factorial prime, a!−1 is ,3,4,6,7,12,14,30,32,33,38,94,166,324,379,469,546,974,
A002981 factorial prime, a!+1 is ,0,1,2,3,11,27,37,41,73,77,116,154,320,340,399,427,872,
A019434 Fermat primes; primes of form 22k+1 ,3,5,17,257,
A024675 interprimes; averages of 2 consecutive odd primes ,4,6,9,12,15,18,21,26,30,34,39,42,45,50,56,60,64,69,72,76,81,86,93,99,102,105,
A051650 lonely numbers; distance to closest prime sets a record ,0,23,53,120,211,
A023186 lonely/isolated primes; record distance to nearest prime ,2,5,23,53,211,
A001348 Mersenne numbers; 2p−1 ,3,7,31,127,
A000668 Mersenne primes; primes of form 2p−1 ,3,7,31,127,
A000043 exponents of Mersenne primes; primes p such that 2p−1 is prime ,2,3,5,7,13,17,19,31,61,89,107,127,521,607,
A016027 indexes of Mersenne primes; 2prime(a)−1 is prime ,1,2,3,4,6,7,8,11,18,24,28,31,98,111,207,328,339,455,583,602,
A007053 number of primes ≤ 2n ,0,1,2,4,6,11,18,31,54,97,172,309,564,
A064403 prime(a)−a and prime(a)+a are primes ,4,6,18,42,66,144,282,384,408,450,522,564,618,672,720,732,744,828,858,
A064269 prime(a)−a is prime ,3,4,6,8,10,14,16,18,28,30,42,44,50,54,66,68,76,84,90,94,110,144,148,154,168,
A064402 prime(a)+a is prime ,1,2,4,6,18,22,24,26,32,34,42,48,66,70,72,82,92,96,98,100,102,104,106,108,114,
A002386 primes with record gaps to following prime ,2,3,7,23,89,113,523,887,
A000101 primes with record gaps to preceding prime ,3,5,11,29,97,127,541,907,
A000230 primes, an = smallest p such that (next prime after p) − p = 2n ,2,3,7,23,89,139,199,113,1831,523,887,1129,1669,2477,2971,4297,5591,1327,9551,
A353074 numbers that differ from their prime neighbours by distinct squares ,140,148,182,190,242,250,284,292,338,346,410,418,422,430,548,556,578,586,632,
A001359 twin primes, lesser of ,3,5,11,17,29,41,59,71,101,107,137,149,179,191,197,227,239,269,281,311,347,
A014574 twin primes, average of ,4,6,12,18,30,42,60,72,102,108,138,150,180,192,198,228,240,270,282,312,348,
A006512 twin primes, greater of ,5,7,13,19,31,43,61,73,103,109,139,151,181,193,199,229,241,271,283,313,349,
A208572 smallest twin prime >2n ,3,5,11,17,41,71,137,269,521,
A023200 cousin primes, lesser of ,3,7,13,19,37,43,67,79,97,103,109,127,163,193,223,229,277,307,313,349,379,397,
A087679 cousin primes, average of ,5,9,15,21,39,45,69,81,99,105,111,129,165,195,225,231,279,309,315,351,381,399,
A031505 cousin primes, greater of ,11,17,23,41,47,71,83,101,107,113,131,167,197,227,233,281,311,317,353,383,401,
A006094 products of 2 consecutive primes ,6,15,35,77,143,221,323,437,667,899,
A046301 products of 3 consecutive primes ,30,105,385,
A046302 products of 4 consecutive primes ,210,
A002110 products of first n primes ,1,2,6,30,210,
A002144 Pythagorean primes; primes of form 4k+1 ,5,13,17,29,37,41,53,61,73,89,97,101,109,113,137,149,157,173,181,193,197,229,
A005385 safe primes; primes of form 2p+1 ,5,7,11,23,47,59,83,107,167,179,227,263,347,359,383,467,479,503,563,587,719,
A068873 smallest prime which is a sum of n distinct primes ,2,5,19,17,43,41,79,83,127,131,199,197,283,281,379,389,499,509,643,641,809,
A005384 Sophie Germain primes; primes of form (p−1)/2 ,2,3,5,11,23,29,41,53,83,89,113,131,173,179,191,233,239,251,281,293,359,419,
A050918 Woodall primes; primes of form k×2k−1 ,7,23,383,
A002234 Woodall number a×2a−1 is prime ,2,3,6,30,75,81,115,123,249,362,384,462,512,751,822,
A014234 largest prime ≤2n ,2,3,7,13,31,61,127,251,509,
A014210 next prime after 2n ,2,3,5,11,17,37,67,131,257,521,
A048744 2a−a is prime ,2,3,9,13,19,21,55,261,
A075190 a2 is an interprime ,2,3,8,9,12,15,18,21,25,33,41,51,60,64,72,78,92,112,117,129,138,140,159,165,
A075191 a3 is an interprime ,4,12,16,26,28,36,48,58,66,68,74,78,102,106,112,117,124,126,129,130,148,152,
A173037 a−4, a−2, a+2 and a+4 are primes ,9,15,105,195,825,
A056809 a, a+1 and a+2 are semiprimes ,33,85,93,121,141,201,213,217,301,393,445,633,697,841,921,

Divisors

OEIS Description First terms
A000005 tau(n); number of divisors of n ,1,2,2,3,2,4,2,4,3,4,2,6,2,4,4,5,2,6,2,6,4,4,2,8,3,4,4,6,2,8,2,6,4,4,4,9,2,4,
A000203 sigma(n); sum of divisors of n ,1,3,4,7,6,12,8,15,13,18,12,28,14,24,24,31,18,39,20,42,32,36,24,60,31,42,40,
A001065 sigma(n)−n; sum of proper divisors of n ,0,1,1,3,1,6,1,7,4,8,1,16,1,10,9,15,1,21,1,22,11,14,1,36,6,16,13,28,1,42,1,31,
A007955 product of divisors of n ,1,2,3,8,5,36,7,64,27,100,11,1728,13,196,225,1024,17,5832,19,8000,441,484,23,
A007956 product of proper divisors of n ,1,1,1,2,1,6,1,8,3,10,1,144,1,14,15,64,1,324,1,400,21,22,1,13824,5,26,27,784,
A005100 sigma(a)<2a; deficient numbers ,1,2,3,4,5,7,8,9,10,11,13,14,15,16,17,19,21,22,23,25,26,27,29,31,32,33,34,35,
A000396 sigma(a)=2a; perfect numbers ,6,28,496,
A005101 sigma(a)>2a; abundant numbers ,12,18,20,24,30,36,40,42,48,54,56,60,66,70,72,78,80,84,88,90,96,100,102,104,
A005835 a = sum of any subset of its proper divisors; pseudoperfect/semiperfect numbers ,6,12,18,20,24,28,30,36,40,42,48,54,56,60,66,72,78,80,84,88,90,96,100,102,104,
A002182 tau(a)>tau(k) for all k<a; highly composite numbers ,1,2,4,6,12,24,36,48,60,120,180,240,360,720,840,
A002093 sigma(a)>sigma(k) for all k<a; highly abundant numbers ,1,2,3,4,6,8,10,12,16,18,20,24,30,36,42,48,60,72,84,90,96,108,120,144,168,180,
A034090 sigma(a)−a>sigma(k)−k for all k<a ,1,2,4,6,8,10,12,18,20,24,30,36,48,60,72,84,90,96,108,120,144,168,180,216,240,
A034287 product of divisors is greater than that of any smaller number ,1,2,3,4,6,8,10,12,18,20,24,30,36,48,60,72,84,90,96,108,120,168,180,240,336,
A034288 product of proper divisors is greater than that of any smaller number ,1,4,6,8,10,12,18,20,24,30,36,48,60,72,84,90,96,108,120,168,180,240,336,360,
A033950 tau(a) divides a; refactorable/tau numbers ,1,2,8,9,12,18,24,36,40,56,60,72,80,84,88,96,104,108,128,132,136,152,156,180,
A007691 a divides sigma(a); multiply-perfect numbers ,1,6,28,120,496,672,
A005179 smallest a such that tau(a)=n ,1,2,4,6,16,12,64,24,36,48,1024,60,4096,192,144,120,65536,180,262144,240,576,
A051444 smallest a such that sigma(a)=n ,1,0,2,3,0,5,4,7,0,0,0,6,9,13,8,0,0,10,0,19,0,0,0,14,0,0,0,12,0,29,16,21,0,0,
A070015 smallest a such that sigma(a)−a=n ,2,0,4,9,0,6,8,10,15,14,21,121,27,22,16,12,39,289,65,34,18,20,57,529,95,46,69,
A006218 tau(1)+…+tau(n) ,0,1,3,5,8,10,14,16,20,23,27,29,35,37,41,45,50,52,58,60,66,70,74,76,84,87,91,
A024916 sigma(1)+…+sigma(n) ,1,4,8,15,21,33,41,56,69,87,99,127,141,165,189,220,238,277,297,339,371,407,
A153485 sigma(1)−1+…+sigma(n)−n ,0,1,2,5,6,12,13,20,24,32,33,49,50,60,69,84,85,106,107,129,140,154,155,191,
A006532 sigma(a) is a square ,1,3,22,66,70,81,94,115,119,170,210,214,217,265,282,310,322,343,345,357,364,
A003624 a is relatively prime to sigma(a) and a composite number; Duffinian numbers ,4,8,9,16,21,25,27,32,35,36,39,49,50,55,57,63,64,65,75,77,81,85,93,98,100,111,
A054973 number of numbers k such that sigma(k)=n ,1,0,1,1,0,1,1,1,0,0,0,2,1,1,1,0,0,2,0,1,0,0,0,3,0,0,0,1,0,1,2,2,0,0,0,1,0,1,
A048138 number of numbers k such that sigma(k)−k=n ,0,1,1,0,2,1,2,1,1,1,1,2,2,2,2,2,1,2,2,3,2,2,1,3,1,2,1,2,1,5,2,3,1,3,1,4,1,1,
A138171 tau(a)>tau(a+1) and a is odd ,45,81,105,117,165,225,261,273,297,315,325,333,345,357,385,405,435,441,465,
A067828 sigma(a)>sigma(a+1) and a is odd ,45,105,117,165,225,273,297,315,345,357,405,465,513,525,561,585,621,693,705,
A206026 smallest a such that sigma(k)=a with at least n values of k ,1,12,24,72,72,168,240,336,360,504,576,720,720,720,720,
A007368 smallest a such that sigma(k)=a with exactly n values of k ,2,1,12,24,96,72,168,240,336,360,504,576,1512,1080,1008,720,2304,3600,5376,
A123930 smallest a such that sigma(k)−k=a with at least n values of k ,2,3,6,21,31,31,49,73,73,91,115,121,121,121,169,169,211,211,211,211,211,301,
A125601 smallest a such that sigma(k)−k=a with exactly n values of k ,2,3,6,21,37,31,49,79,73,91,115,127,151,121,181,169,217,265,253,271,211,301,
A007369 sigma(k)=a with no number k ,2,5,9,10,11,16,17,19,21,22,23,25,26,27,29,33,34,35,37,41,43,45,46,47,49,50,
A145899 sigma(k)=a with more numbers k than for any smaller a ,1,12,24,72,168,240,336,360,504,576,720,
A005114 sigma(k)−k=a with no number k; untouchable numbers ,2,5,52,88,96,120,124,146,162,188,206,210,216,238,246,248,262,268,276,288,290,
A238895 sigma(k)−k=a with more numbers k than for any smaller a ,2,3,6,21,31,49,73,91,115,121,169,211,301,331,361,391,421,511,631,721,781,841,
A002025 a=sigma(k)−k and k=sigma(a)−a for any k>a; smaller of an amicable pair ,220,
A002046 a=sigma(k)−k and k=sigma(a)−a for any k<a; larger of an amicable pair ,284,
A006037 abundant but not pseudoperfect numbers; weird numbers ,70,836,
A083207 divisors can be partitioned into two sets with equal sum; Zumkeller numbers ,6,12,20,24,28,30,40,42,48,54,56,60,66,70,78,80,84,88,90,96,102,104,108,112,
A076985 smallest Fibonacci number having exactly n Fibonacci divisors ,1,2,8,610,144,1134903170,46368,14930352,4807526976,

Sums and differences (not primes)

OEIS Description First terms
A053767 sums of first n composite numbers ,0,4,10,18,27,37,49,63,78,94,112,132,153,175,199,224,250,277,305,335,367,400,
A051349 sums of first n nonprimes ,0,1,5,11,19,28,38,50,64,79,95,113,133,154,176,200,225,251,278,306,336,368,
A014439 differences of 2 positive cubes, in 1 way ,7,19,26,37,56,61,63,91,98,117,124,127,152,169,189,208,215,217,218,271,279,
A034179 differences of 2 positive cubes, in 2+ ways ,721,728,999,
A003108 number of partitions of n into cubes ,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,5,5,5,5,5,6,6,6,7,7,7,
A279329 number of partitions of n into distinct cubes ,1,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,1,0,
A338667 sums of 2 positive cubes in 1 way ,2,9,16,28,35,54,65,72,91,126,128,133,152,189,217,224,243,250,280,341,344,351,
A001235 sums of 2 positive cubes in 2+ ways; taxicab numbers (none below 1000)
A024670 cubes, sums of 2 distinct positive ,9,28,35,65,72,91,126,133,152,189,217,224,243,280,341,344,351,370,407,468,513,
A025395 cubes, sums of 3 positive, in 1 way ,3,10,17,24,29,36,43,55,62,66,73,80,81,92,99,118,127,129,134,136,141,153,155,
A008917 cubes, sums of 3 positive, in 2+ ways ,251,
A024975 cubes, sums of 3 distinct positive ,36,73,92,99,134,153,160,190,197,216,225,244,251,281,288,307,342,349,352,368,
A025403 cubes, sums of 4 positive, in 1 way ,4,11,18,25,30,32,37,44,51,56,63,67,70,74,81,82,88,89,93,100,107,108,119,126,
A025406 cubes, sums of 4 positive, in 2+ ways ,219,252,259,278,315,376,467,522,594,702,758,763,765,802,809,819,856,864,945,
A025411 cubes, sums of 4 distinct positive ,100,161,198,217,224,252,289,308,315,350,369,376,379,406,413,416,432,435,442,
A018888 cubes, sums of 7 or fewer positive, in 0 ways (full list) ,15,22,23,50,114,167,175,186,212,231,238,239,303,364,420,428,454,
A001476 cubes, sums of distinct positive, in 0 ways ,2,3,4,5,6,7,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,29,30,31,32,
A003997 cubes, sums of distinct positive, in 1+ ways ,1,8,9,27,28,35,36,64,65,72,73,91,92,99,100,125,126,133,134,152,153,160,161,
A000537 cubes, sums of first n positive; squares of triangular numbers ,0,1,9,36,100,225,441,784,
A274132 cube, sum of 3 positive, an is for all positive n ,134,153,216,225,244,251,288,342,368,405,408,415,528,532,540,577,645,729,750,
A076980 Leyland numbers, 1st kind; bc+cb ,3,8,17,32,54,57,100,145,177,320,368,512,593,945,
A045575 Leyland numbers, 2nd kind; bc−cb ,0,1,7,17,28,79,118,192,399,431,513,924,
A007925 nn+1−(n+1)n -1,-1,-1,17,399,
A024352 squares, differences of 2 positive ,3,5,7,8,9,11,12,13,15,16,17,19,20,21,23,24,25,27,28,29,31,32,33,35,36,37,39,
A306102 squares, differences of 2 positive, in 2+ ways ,15,21,24,27,32,33,35,39,40,45,48,51,55,56,57,60,63,64,65,69,72,75,77,80,81,
A306103 squares, differences of 2 positive, in 3+ ways ,45,48,63,72,75,80,96,99,105,112,117,120,128,135,144,147,153,160,165,168,171,
A306104 squares, differences of 2 positive, in 4+ ways ,96,105,120,135,144,160,165,168,189,192,195,216,224,225,231,240,255,264,273,
A100073 squares, differences of 2 positive, number of representations of n ,0,0,1,0,1,0,1,1,1,0,1,1,1,0,2,1,1,0,1,1,2,0,1,2,1,0,2,1,1,0,1,2,2,0,2,1,1,0,
A334078 squares, differences of 2 positive, smallest in at least n ways ,3,15,45,96,192,240,480,480,720,960,
A094191 squares, differences of 2 positive, smallest in exactly n ways ,3,15,45,96,192,240,576,480,720,960,12288,1440,3600,3840,2880,3360,20736,5040,
A025426 squares, number of partitions of n into 2 ,0,0,1,0,0,1,0,0,1,0,1,0,0,1,0,0,0,1,1,0,1,0,0,0,0,1,1,0,0,1,0,0,1,0,1,0,0,1,
A025441 squares, number of partitions of n into 2 distinct ,0,0,0,0,0,1,0,0,0,0,1,0,0,1,0,0,0,1,0,0,1,0,0,0,0,1,1,0,0,1,0,0,0,0,1,0,0,1,
A025427 squares, number of partitions of n into 3 ,0,0,0,1,0,0,1,0,0,1,0,1,1,0,1,0,0,1,1,1,0,1,1,0,1,0,1,2,0,1,1,0,0,2,1,1,1,0,
A025442 squares, number of partitions of n into 3 distinct ,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,1,0,0,1,1,0,0,0,0,1,0,0,
A025428 squares, number of partitions of n into 4 ,0,0,0,0,1,0,0,1,0,0,1,0,1,1,0,1,1,0,1,1,1,1,1,1,0,1,1,1,3,0,1,2,0,1,2,1,2,2,
A025443 squares, number of partitions of n into 4 distinct ,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,
A001156 squares, number of partitions of n into ,1,1,1,1,2,2,2,2,3,4,4,4,5,6,6,6,8,9,10,10,12,13,14,14,16,19,20,21,23,26,27,
A033461 squares, number of partitions of n into distinct ,1,1,0,0,1,1,0,0,0,1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0,2,2,0,0,2,2,0,0,0,1,1,1,1,
A025284 squares, sums of 2, in 1 way ,2,5,8,10,13,17,18,20,25,26,29,32,34,37,40,41,45,52,53,58,61,68,72,73,74,80,
A025285 squares, sums of 2, in 2 ways ,50,65,85,125,130,145,170,185,200,205,221,250,260,265,290,305,338,340,365,370,
A025294 squares, sums of 2, in 3+ ways ,325,425,650,725,845,850,925,
A025302 squares, sums of 2 distinct, in 1 way ,5,10,13,17,20,25,26,29,34,37,40,41,45,50,52,53,58,61,68,73,74,80,82,89,90,97,
A025303 squares, sums of 2 distinct, in 2 ways ,65,85,125,130,145,170,185,205,221,250,260,265,290,305,340,365,370,377,410,
A025313 squares, sums of 2 distinct, in 3+ ways ,325,425,650,725,845,850,925,
A025321 squares, sums of 3, in 1 way ,3,6,9,11,12,14,17,18,19,21,22,24,26,29,30,34,35,36,42,43,44,45,46,48,49,50,
A025322 squares, sums of 3, in 2 ways ,27,33,38,41,51,57,59,62,69,74,75,77,83,90,94,98,102,105,107,108,113,117,118,
A025323 squares, sums of 3, in 3 ways ,54,66,81,86,89,99,101,110,114,126,131,149,150,162,166,173,174,179,182,185,
A025324 squares, sums of 3, in 4 ways ,129,134,146,153,161,171,189,198,201,234,243,246,249,251,254,257,261,270,278,
A025333 squares, sums of 3, in 5+ ways ,194,206,209,230,266,269,281,297,306,314,321,326,329,341,342,350,354,369,374,
A025339 squares, sums of 3 distinct, in 1 way ,14,21,26,29,30,35,38,41,42,45,46,49,50,53,54,56,59,61,65,66,70,75,78,81,83,
A025340 squares, sums of 3 distinct, in 2 ways ,62,69,74,77,86,89,90,94,98,105,117,122,125,129,131,138,141,150,154,155,158,
A025341 squares, sums of 3 distinct, in 3 ways ,101,110,126,134,146,149,173,174,182,185,186,221,222,237,245,251,257,278,286,
A025342 squares, sums of 3 distinct, in 4 ways ,161,189,194,209,234,254,261,270,281,285,290,293,299,321,362,365,369,371,378,
A025351 squares, sums of 3 distinct, in 5+ ways ,206,230,266,269,314,326,329,341,350,374,381,389,398,413,414,425,426,434,437,
A120328 squares, sums of 3 consecutive ,2,5,14,29,50,77,110,149,194,245,302,365,434,509,590,677,770,869,974,
A025357 squares, sums of 4, in 1 way ,4,7,10,12,13,15,16,18,19,20,21,22,23,25,26,27,30,33,35,38,40,44,46,48,51,53,
A025358 squares, sums of 4, in 2 ways ,31,34,36,37,39,43,45,47,49,50,54,57,61,68,69,71,74,77,81,83,86,94,107,113,
A025359 squares, sums of 4, in 3 ways ,28,42,55,60,66,67,73,75,78,79,85,92,95,99,109,110,112,121,125,129,134,137,
A025360 squares, sums of 4, in 4 ways ,52,58,63,70,76,84,87,91,93,97,98,105,119,123,139,140,141,142,146,155,158,185,
A025361 squares, sums of 4, in 5 ways ,82,100,102,103,106,108,111,114,115,117,118,122,126,127,132,143,145,151,153,
A025362 squares, sums of 4, in 6 ways ,90,124,133,147,156,157,159,163,165,166,171,174,177,188,193,201,203,205,219,
A025363 squares, sums of 4, in 7 ways ,135,148,170,172,182,183,187,189,190,199,215,229,245,261,263,289,305,317,347,
A025364 squares, sums of 4, in 8 ways ,130,138,150,154,175,180,186,195,196,213,214,217,218,222,228,230,235,237,238,
A025365 squares, sums of 4, in 9 ways ,162,178,207,220,223,225,226,231,242,243,253,265,266,267,271,278,283,286,287,
A025375 squares, sums of 4, in 10+ ways ,198,202,210,234,246,247,250,252,255,258,262,268,270,273,274,279,282,285,290,
A025376 squares, sums of 4 distinct, in 1 way ,30,39,46,50,51,54,57,62,63,65,66,70,71,74,75,79,81,84,85,86,87,91,93,98,106,
A025377 squares, sums of 4 distinct, in 2 ways ,90,94,95,99,105,111,119,123,129,134,138,141,143,146,151,153,154,155,166,167,
A025378 squares, sums of 4 distinct, in 3 ways ,78,102,110,114,130,135,147,156,159,171,175,177,189,191,194,201,204,205,211,
A025379 squares, sums of 4 distinct, in 4 ways ,142,158,162,165,182,183,195,206,207,214,215,218,226,239,243,245,259,260,262,
A025380 squares, sums of 4 distinct, in 5 ways ,126,150,170,186,219,225,230,242,249,250,261,267,274,275,278,287,295,297,305,
A025390 squares, sums of 4 distinct, in 6+ ways ,174,190,198,210,222,231,234,238,246,254,255,258,266,270,273,279,282,285,286,
A027575 squares, sums of 4 consecutive ,14,30,54,86,126,174,230,294,366,446,534,630,734,846,966,
A027578 squares, sums of 5 consecutive ,30,55,90,135,190,255,330,415,510,615,730,855,990,
A001422 squares, sums of distinct, in 0 ways ,2,3,6,7,8,11,12,15,18,19,22,23,24,27,28,31,32,33,43,44,47,48,60,67,72,76,92,
A003995 squares, sums of distinct, in 1+ ways ,0,1,4,5,9,10,13,14,16,17,20,21,25,26,29,30,34,35,36,37,38,39,40,41,42,45,46,
A003996 squares, sums of distinct, in 2+ ways ,25,26,29,30,41,45,46,49,50,53,54,61,62,65,66,69,70,74,75,77,78,79,81,82,84,
A097563 squares, sums of distinct, least number in exactly n ways ,2,0,25,50,65,94,90,110,155,126,191,170,186,174,190,211,195,226,210,231,234,
A097758 squares, sums of distinct, greatest number in exactly n ways ,128,132,188,192,193,213,228,224,253,288,257,293,297,292,317,301,333,284,337,
A078135 numbers that cannot be written as a sum of squares greater than 1 (full list) ,1,2,3,5,6,7,10,11,14,15,19,23,
A078360 sums of a cube and a square in 1 way ,2,5,9,10,12,24,26,28,31,33,36,37,43,44,50,52,57,63,68,72,73,76,80,82,91,100,
A054402 sums of a cube and a square in 2+ ways ,17,65,89,108,129,145,225,233,252,297,316,388,449,464,505,537,548,577,593,633,
A171385 sums of a cube and a square in 3+ ways (none below 1000)
A007294 triangular numbers, number of partitions of n into positive ,1,1,1,2,2,2,4,4,4,6,7,7,10,11,11,15,17,17,22,24,25,32,35,36,44,48,50,60,66,
A024940 triangular numbers, number of partitions of n into positive distinct ,1,1,0,1,1,0,1,1,0,1,2,1,0,1,1,1,2,1,1,2,1,2,2,0,2,3,1,1,3,2,1,4,3,0,3,3,2,4,
A053614 triangular numbers, a is not sum of distinct (complete list) ,2,5,8,12,23,33,
A051533 triangular numbers, sums of 2 positive ,2,4,6,7,9,11,12,13,16,18,20,21,22,24,25,27,29,30,31,34,36,37,38,39,42,43,46,
A265140 triangular numbers, sums of 2 positive distinct, in 1 way ,4,7,9,11,13,18,21,22,24,25,27,29,34,36,37,38,39,42,43,48,49,55,56,57,58,60,
A265134 triangular numbers, sums of 2 positive distinct, in 2 ways ,16,31,46,51,76,94,111,121,123,126,133,141,146,156,157,172,174,186,191,196,
A265136 triangular numbers, sums of 2 positive distinct, in 3 ways ,81,106,181,211,256,276,331,361,381,406,456,556,606,631,666,681,706,718,731,
A265137 triangular numbers, sums of 2 positive distinct, in 4+ ways ,471,531,601,616,786,871,906,991,

Formulas

OEIS Description First terms
A005245 number of 1's required to build n using + * ,1,2,3,4,5,5,6,6,6,7,8,7,8,8,8,8,9,8,9,9,9,10,11,9,10,10,9,10,11,10,11,10,11,
A025280 number of 1's required to build n using + * ^ ,1,2,3,4,5,5,6,5,5,6,7,7,8,8,8,6,7,7,8,8,9,9,10,8,7,8,6,7,8,9,10,7,8,9,10,7,8,
A091333 number of 1's required to build n using + − * () ,1,2,3,4,5,5,6,6,6,7,8,7,8,8,8,8,9,8,9,9,9,10,10,9,10,10,9,10,11,10,11,10,11,
A091334 number of 1's required to build n using + − * ^ () ,1,2,3,4,5,5,6,5,5,6,7,7,8,8,7,6,7,7,8,8,9,9,9,8,7,7,6,7,8,9,8,7,8,9,8,7,8,9,
A329526 number of 1's required to build n using + − * ^ ! ,1,2,3,4,4,3,4,5,5,6,6,5,6,6,7,6,7,6,7,7,7,6,5,4,5,6,6,7,8,7,7,6,7,7,6,5,6,7,
A348262 number of 1's required to build n using + ^ ,1,2,3,4,5,6,7,5,5,6,7,8,9,10,11,6,7,8,9,10,11,12,13,11,7,8,6,7,8,9,10,7,8,9,
A378758 number of 1's required to build n using + − ^ ,1,2,3,4,5,6,6,5,5,6,7,8,9,8,7,6,7,8,9,10,11,10,9,8,7,7,6,7,8,9,8,7,8,9,9,8,9,
A378759 number of 1's required to build n using + / ^ ,1,2,3,4,5,6,7,5,5,6,7,8,9,9,10,6,7,8,9,10,11,12,13,11,7,8,6,7,8,9,10,7,8,9,
A005520 smallest number requiring n 1's to build using + * ,1,2,3,4,5,7,10,11,17,22,23,41,47,59,89,107,167,179,263,347,467,683,719,
A255641 smallest number requiring n 1's to build using + − * ,1,2,3,4,5,7,10,11,17,22,29,41,58,67,101,131,173,262,346,461,617,787,
A003037 smallest number requiring n 1's to build using + * ^ ,1,2,3,4,5,7,11,13,21,23,41,43,71,94,139,211,215,431,863,
A347983 smallest number requiring n 1's to build using + − * ^ ,1,2,3,4,5,7,11,13,21,39,41,43,115,173,276,413,823,
A253177 numbers that can be built from fewer 1's using − in addition to + * ,23,47,53,59,69,71,89,94,106,107,134,141,142,143,159,161,167,177,178,179,188,
A348069 numbers that can be built from fewer 1's using / in addition to + − * (none below 1000)
A213923 minimal lengths of formulas representing n using 1 + * ,1,3,5,7,9,9,11,11,11,13,15,13,15,15,15,15,17,15,17,17,17,19,21,17,19,19,17,
A217250 minimal lengths of formulas representing n using 1 + * ^ ,1,3,5,7,9,9,11,9,9,11,13,13,15,15,15,11,13,13,15,15,17,17,19,15,13,15,11,13,
A213924 minimal lengths of formulas representing n using 1 + ^ ,1,3,5,7,9,11,13,9,9,11,13,15,17,19,21,11,13,15,17,19,21,23,25,21,13,15,11,13,
A182002 smallest positive integer that cannot be computed using exactly n n's and + − * / () ,2,2,1,10,13,22,38,91,195,443,634,
A181957 smallest positive integer that cannot be computed using n operators (+ *) with integer operands 1–9 and () ,10,19,92,239,829,
A181898 smallest positive integer that cannot be computed using n operators (+ − * /) with integer operands 1–9 and () ,10,19,92,417,851,
A005208 number of operators (+ * ^) needed to build n from 1's ,0,1,2,3,4,4,5,4,4,5,6,6,7,7,7,5,6,6,7,7,8,8,9,7,6,7,5,6,7,8,9,6,7,8,9,6,7,8,

Geometry

OEIS Description First terms
A000124 maximum number of pieces formed by slicing a disk with n cuts; lazy caterer's sequence ,1,2,4,7,11,16,22,29,37,46,56,67,79,92,106,121,137,154,172,191,211,232,254,
A000125 maximum number of pieces formed by slicing a cube with n cuts; cake numbers ,1,2,4,8,15,26,42,64,93,130,176,232,299,378,470,576,697,834,988,
A003600 maximum number of pieces formed by slicing a torus with n cuts ,1,2,6,13,24,40,62,91,128,174,230,297,376,468,574,695,832,986,
A092536 polyhedra, Archimedean, number of edges ,18,24,36,36,48,60,60,72,90,90,120,150,180,
A092537 polyhedra, Archimedean, number of faces ,8,14,14,14,26,26,32,32,32,38,62,62,92,
A092538 polyhedra, Archimedean, number of vertices ,12,12,24,24,24,24,30,48,60,60,60,60,120,
A063722 polyhedra, Platonic, number of edges ,6,12,12,30,30,
A053016 polyhedra, Platonic, number of faces and vertices ,4,6,8,12,20,
A257479 maximal kissing number in n dimensions ,2,6,12,24,
A353935 a cube cannot be divided into a subcubes ,2,3,4,5,6,7,9,10,11,12,13,14,16,17,18,19,21,23,24,25,26,28,30,31,32,33,35,37,
A003401 regular a-gon is constructible with ruler & compass ,1,2,3,4,5,6,8,10,12,15,16,17,20,24,30,32,34,40,48,51,60,64,68,80,85,96,102,
A000105 number of free polyominoes with n cells ,1,1,1,2,5,12,35,108,369,
A000104 number of free polyominoes with n cells and no holes ,1,1,1,2,5,12,35,107,363,
A001419 number of free polyominoes with n cells and at least one hole ,0,0,0,0,0,0,1,6,37,195,979,
A000988 number of one-sided polyominoes with n cells ,1,1,1,2,7,18,60,196,704,
A001168 number of fixed polyominoes with n cells ,1,1,2,6,19,63,216,760,

Misc

OEIS Description First terms
A003101 1k + 2k−1 + … + (k−1)2 + k1 ,0,1,3,8,22,65,209,732,
A000312 aa ,1,1,4,27,256,
A000110 Bell numbers; number of ways to partition a set of n labeled elements ,1,1,2,5,15,52,203,877,
A000108 Catalan numbers ,1,1,2,5,14,42,132,429,
A002808 composite numbers; mn for m, n > 1 ,4,6,8,9,10,12,14,15,16,18,20,21,22,24,25,26,27,28,30,32,33,34,35,36,38,39,40,
A013921 composite numbers equal to the sum of the first k composites for some k ,4,10,18,27,49,63,78,94,112,132,153,175,224,250,305,335,400,434,469,505,543,
A053781 composite numbers, a divides the sum of the first a ,1,2,3,7,11,71,107,115,139,155,681,
A002064 Cullen numbers; n×2n+1 ,1,3,9,25,65,161,385,897,
A059756 Erdős–Woods numbers ,16,22,34,36,46,56,64,66,70,76,78,86,88,92,94,96,100,106,112,116,118,120,124,
A000142 factorial numbers; n! ,1,1,2,6,24,120,720,
A000045 Fibonacci numbers; a0=0, a1=1, an=an−1+an−2 ,0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,
A023172 Self-Fibonacci numbers; a divides Fibonacci(a) ,1,5,12,24,25,36,48,60,72,96,108,120,125,144,168,180,192,216,240,288,300,324,
A088959 hypotenuses of Pythagorean triangles, in more ways than any smaller number ,1,5,25,65,325,
A006339 hypotenuses of Pythagorean triangles, least of exactly n ,1,5,25,125,65,3125,15625,325,390625,1953125,1625,48828125,4225,1105,
A084645 hypotenuses of Pythagorean triangles, of 1 (multiples of Pythagorean primes) ,5,10,13,15,17,20,26,29,30,34,35,37,39,40,41,45,51,52,53,55,58,60,61,68,70,73,
A084646 hypotenuses of Pythagorean triangles, of 2 ,25,50,75,100,150,169,175,200,225,275,289,300,338,350,400,450,475,507,525,550,
A084648 hypotenuses of Pythagorean triangles, of 4 ,65,85,130,145,170,185,195,205,221,255,260,265,290,305,340,365,370,377,390,
A008846 hypotenuses of primitive Pythagorean triangles (products of Pythagorean primes) ,5,13,17,25,29,37,41,53,61,65,73,85,89,97,101,109,113,125,137,145,149,157,169,
A024409 hypotenuses of primitive Pythagorean triangles, of 2+ ,65,85,145,185,205,221,265,305,325,365,377,425,445,481,485,493,505,533,545,
A159781 hypotenuses of primitive Pythagorean triangles, of 4 (none below 1000)
A000032 Lucas numbers; a0=2, a1=1, an=an−1+an−2 ,2,1,3,4,7,11,18,29,47,76,123,199,322,521,843,
A000959 lucky numbers ,1,3,7,9,13,15,21,25,31,33,37,43,49,51,63,67,69,73,75,79,87,93,99,105,111,115,
A006003 magic constants for n×n magic squares ,0,1,5,15,34,65,111,175,260,369,505,671,870,
A001006 Motzkin numbers ,1,1,2,4,9,21,51,127,323,835,
A000930 Narayana's cows; a0=a1=a2=1, an=an−1+an−3 ,1,1,1,2,3,4,6,9,13,19,28,41,60,88,129,189,277,406,595,872,
A072843 O'Halloran numbers; even and can't be the area of a cuboid with integer sides (full list) ,8,12,20,36,44,60,84,116,140,156,204,260,380,420,660,924,
A000931 Padovan sequence; a0=1, a1=a2=0, an=an−2+an−3 ,1,0,0,1,0,1,1,1,2,2,3,4,5,7,9,12,16,21,28,37,49,65,86,114,151,200,265,351,
A000041 partition numbers ,1,1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,231,297,385,490,627,792,
A000129 Pell numbers; denominators of rational approximations of √2 ,0,1,2,5,12,29,70,169,408,985,
A001608 Perrin sequence; a0=3, a1=0, a2=2, an=an−2+an−3 ,3,0,2,3,2,5,5,7,10,12,17,22,29,39,51,68,90,119,158,209,277,367,486,644,853,
A005153 practical/panarithmic numbers; every k ≤ sigma(a) is a sum of distinct divisors of a ,1,2,4,6,8,12,16,18,20,24,28,30,32,36,40,42,48,54,56,60,64,66,72,78,80,84,88,
A002378 products of 2 consecutive nonnegative integers; oblong/promic/pronic numbers ,0,2,6,12,20,30,42,56,72,90,110,132,156,182,210,240,272,306,342,380,420,462,
A053143 smallest square divisible by n ,1,4,9,4,25,36,49,16,9,100,121,36,169,196,225,16,289,36,361,100,441,484,529,
A000058 Sylvester's sequence; a0=2, an=an−1(an−1−1)+1 ,2,3,7,43,
A002858 Ulam numbers ,1,2,3,4,6,8,11,13,16,18,26,28,36,38,47,48,53,57,62,69,72,77,82,87,97,99,102,
A003261 Woodall/Riesel numbers; n×2n−1 ,1,7,23,63,159,383,895,

Radix-dependent

These are only interesting when represented in the specified base or involve reversing digits in the specified base.

Decimal – reversals and permutations

OEIS Description First terms
A057708 2a is a prime reversed ,1,4,5,7,10,17,24,37,45,55,70,77,107,137,150,271,364,
A379938 prime(a) is a power of two reversed ,1,9,18,142,575,
A036971 a'th Fibonacci number is a prime reversed ,3,4,5,7,9,14,17,21,25,26,65,98,175,191,382,497,653,
A036972 prime(a) is a Fibonacci number reversed ,1,2,3,11,14,137,
A007488 primes whose reversal is a square ,61,163,487,691,
A068989 squares whose reversal is prime ,16,196,361,784,
A370071 Fibonacci numbers that are a permutation of a perfect power ,0,1,8,144,610,

Decimal – primes

OEIS Description First terms
A002385 palindromic primes ,2,3,5,7,11,101,131,151,181,191,313,353,373,383,727,757,787,797,919,929,
A006567 emirps; a prime and a different prime backwards ,13,17,31,37,71,73,79,97,107,113,149,157,167,179,199,311,337,347,359,389,701,
A161851 a square and an anagram of a prime ,16,121,169,196,289,361,784,961,
A161853 a prime and an anagram of a square ,61,163,211,487,613,619,631,691,829,
A161855 a cube and an anagram of a prime ,125,343,512,
A161854 a prime and an anagram of a cube ,251,433,521,
A003459 absolute/permutable primes; every permutation is prime ,2,3,5,7,11,13,17,31,37,71,73,79,97,113,131,199,311,337,373,733,919,991,
A052015 primes with distinct digits in ascending order ,2,3,5,7,13,17,19,23,29,37,47,59,67,79,89,127,137,139,149,157,167,179,239,257,
A052014 primes with distinct digits in descending order ,2,3,5,7,31,41,43,53,61,71,73,83,97,421,431,521,541,631,641,643,653,743,751,
A024770 right-truncatable primes; every prefix is prime ,2,3,5,7,23,29,31,37,53,59,71,73,79,233,239,293,311,313,317,373,379,593,599,
A035244 smallest number with exactly n prime substrings ,1,2,13,23,113,137,373,1137,1733,1373,11317,11373,13733,31373,113173,131373,
A001101 Moran numbers; a/(sum of digits) is prime ,18,21,27,42,45,63,84,111,114,117,133,152,153,156,171,190,195,198,201,207,209,
A006753 Smith numbers; sum of digits = sum of digits of prime factors ,4,22,27,58,85,94,121,166,202,265,274,319,346,355,378,382,391,438,454,483,517,
A004023 (10a−1)/9 (a ones) is prime ,2,19,23,317,
A003020 largest prime factor of (10n−1)/9 (n ones) ,11,37,101,271,37,4649,137,333667,9091,513239,9901,265371653,909091,2906161,
A107289 sum of digits of a2 is prime ,4,5,7,16,17,25,26,28,29,32,34,37,38,40,44,47,49,50,52,53,56,61,62,64,65,70,
A076204 sum of digits of a3 is prime ,13,14,16,17,22,23,25,28,34,37,47,52,58,64,67,68,74,76,85,106,107,118,130,134,
A076203 sum of digits of 2a is prime ,1,4,5,7,8,10,12,18,19,20,23,24,25,28,29,30,31,34,35,38,39,40,48,52,63,66,68,
A075177 sum of digits of prime(a) is prime ,1,2,3,4,5,9,10,13,14,15,18,19,23,24,26,30,32,33,34,36,37,40,41,43,44,45,46,

Decimal – number of distinct digits

OEIS Description First terms
A115938 aa contains a pandigital substring ,125,132,147,162,163,167,169,176,186,188,192,197,209,215,218,222,223,237,247,
A137214 number of distinct digits in 2n ,1,1,1,1,2,2,2,3,3,3,4,4,4,4,5,5,3,5,4,4,7,6,5,4,4,4,6,6,6,9,7,7,5,6,6,7,7,8,
A309432 number of distinct digits in n2 ,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,2,3,3,3,3,2,2,2,3,3,3,2,3,3,3,2,3,4,4,3,3,4,4,
A243909 a is the smallest number such that 2a contains exactly n distinct digits ,1,4,7,10,14,21,20,37,29,68,
A054039 a is the smallest number such that a2 contains exactly n distinct digits ,0,4,13,32,113,322,1017,3206,10124,32043,
A247794 a is the smallest number such that a3 contains exactly n distinct digits ,0,3,5,12,22,59,135,289,1018,2326,
A020666 a is the smallest number such that an is pandigital ,1023456789,32043,2326,763,309,159,56,104,49,36,25,15,25,17,17,15,16,7,5,6,6,
A112388 a is the smallest prime such that an is pandigital ,10123457689,101723,5437,2339,1009,257,139,173,83,67,31,29,37,17,17,47,19,7,5,
A240069 nk is pandigital starting at k=a (0 if never) ,0,169,107,85,66,65,62,57,54,0,42,52,38,35,35,43,28,26,45,169,30,25,51,24,30,
A178960 a! is pandigital ,23,27,31,33,34,35,36,37,39,40,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,
A182049 a! is not pandigital (all known terms > 23 shown) ,24,25,26,28,29,30,32,38,41,

Decimal – other

OEIS Description First terms
A004207 a0 = 1, an = sum of digits of all previous terms ,1,1,2,4,8,16,23,28,38,49,62,70,77,91,101,103,107,115,122,127,137,148,161,169,
A048345 an2 is the smallest square containing exactly n 0's ,0,10,320,100,3200,1000,32000,10000,320000,100000,3200000,1000000,32000000,
A048346 an2 is the smallest square containing exactly n 1's ,0,1,11,109,1054,3381,10541,105414,414139,1055041,10252371,78173596,334082481,
A048347 an2 is the smallest square containing exactly n 2's ,5,15,149,1415,4585,14585,105935,364585,3496101,4714045,34964585,149305935,
A048348 an2 is the smallest square containing exactly n 3's ,6,56,586,1156,11547,57735,559769,1197219,6582806,36514844,350903624,
A048349 an2 is the smallest square containing exactly n 4's ,2,12,38,212,2538,6888,66592,210771,2059962,6696592,21081538,209868112,
A048350 an2 is the smallest square containing exactly n 5's ,5,75,235,745,22485,22925,235065,505525,2356384,23569166,227069495,674919666,
A048351 an2 is the smallest square containing exactly n 6's ,4,26,216,1291,5164,68313,163284,785294,3559026,26393686,129099069,254296413,
A048352 an2 is the smallest square containing exactly n 7's ,24,76,424,3576,8819,88924,278874,2116076,8819154,61463576,277450424,
A048353 an2 is the smallest square containing exactly n 8's ,9,83,298,1378,8878,29641,298141,623609,9321378,28072917,94121667,329877083,
A048354 an2 is the smallest square containing exactly n 9's ,3,63,173,1414,17313,53937,138923,953937,3082207,31622764,99849687,301579177,
A048365 an3 is the smallest cube containing exactly n 0's ,1,0,52,10,160,520,100,1600,5200,1000,16000,52000,10000,160000,520000,100000,
A048366 an3 is the smallest cube containing exactly n 1's ,1,11,58,106,671,1041,10058,22598,145981,480765,2359231,10297461,4836178,
A048367 an3 is the smallest cube containing exactly n 2's ,3,28,138,587,612,2824,27654,29603,131468,1312748,1616488,2811574,49629974,
A048368 an3 is the smallest cube containing exactly n 3's ,17,7,179,477,707,6935,15477,44197,535677,693368,2028209,7566137,32215777,
A048369 an3 is the smallest cube containing exactly n 4's ,4,14,114,164,763,3543,17066,13464,163974,757364,3421244,6727219,28902604,
A048370 an3 is the smallest cube containing exactly n 5's ,5,25,136,715,1526,11828,8121,115798,319405,1771087,2179693,11665419,38160335,
A048371 an3 is the smallest cube containing exactly n 6's ,4,55,36,716,1188,4055,13832,18821,190806,1542023,3971816,13881356,55009989,
A048372 an3 is the smallest cube containing exactly n 7's ,3,26,83,173,1983,2953,19753,90643,258999,426859,4255753,13955253,42111153,
A048373 an3 is the smallest cube containing exactly n 8's ,2,42,92,436,942,2402,16942,52942,266192,2018892,3069442,14242355,44559402,
A048374 an3 is the smallest cube containing exactly n 9's ,9,31,99,998,999,7937,9999,99998,99999,996999,999999,6688699,9999999,97609999,
A003226 automorphic numbers; a2 ends with a ,0,1,5,6,25,76,376,625,
A033819 trimorphic numbers; a3 ends with a ,0,1,4,5,6,9,24,25,49,51,75,76,99,125,249,251,375,376,499,501,624,625,749,751,
A119509 all digits are distinct in a2 ,1,2,3,4,5,6,7,8,9,13,14,16,17,18,19,23,24,25,27,28,29,31,32,33,36,37,42,43,
A129525 all digits are distinct in a3 ,1,2,3,4,5,6,8,9,12,13,16,17,18,19,21,22,24,27,29,32,35,38,41,59,66,69,73,75,
A059930 all digits are distinct in a and a2 combined (full list) ,2,3,4,7,8,9,17,18,24,29,53,54,57,59,72,79,84,209,259,567,807,854,
A305734 all digits are distinct in each of a, a2 and a3 (full list) ,0,1,2,3,4,5,6,8,9,13,16,17,18,19,24,27,29,32,59,69,73,84,93,203,289,302,
A030097 all digits are even in a2 ,0,2,8,20,22,68,78,80,92,162,168,200,202,220,262,298,478,492,498,668,680,780,
A030099 all digits are odd in a3 ,1,11,15,33,39,71,91,173,175,179,211,259,335,
A050741 no consecutive equal digits in a2 ,0,1,2,3,4,5,6,7,8,9,11,13,14,16,17,18,19,22,23,24,25,26,27,28,29,31,32,33,36,
A050742 no consecutive equal digits in a3 ,0,1,2,3,4,5,6,7,8,9,12,13,16,17,18,19,21,22,23,24,25,26,27,28,29,31,32,33,34,
A256601 1 is the smallest digit and 9 is the largest in each of a and a2 ,139,219,519,591,719,891,911,961,971,981,
A036057 Friedman numbers; can be written nontrivially using digits and +−×/^ and concatenation ,25,121,125,126,127,128,153,216,289,343,347,625,688,736,
A007532 handsome numbers; sum of positive powers of its digits ,1,2,3,4,5,6,7,8,9,24,43,63,89,132,135,153,175,209,224,226,262,264,267,283,
A007770 happy numbers ,1,7,10,13,19,23,28,31,32,44,49,68,70,79,82,86,91,94,97,100,103,109,129,130,
A005349 Harshad/Niven numbers; a is divisible by the sum of its digits ,1,2,3,4,5,6,7,8,9,10,12,18,20,21,24,27,30,36,40,42,45,48,50,54,60,63,70,72,
A006886 Kaprekar numbers; see OEIS ,1,9,45,55,99,297,703,999,
A115569 Lynch-Bell numbers; divisible by each digit, digits distinct, no 0 ,1,2,3,4,5,6,7,8,9,12,15,24,36,48,124,126,128,132,135,162,168,175,184,216,248,
A003001 smallest number of multiplicative persistence n ,0,10,25,39,77,679,
A068669 noncomposite and every substring is noncomposite ,1,2,3,5,7,11,13,17,23,31,37,53,71,73,113,131,137,173,311,313,317,373,
A002113 palindromes (terms < 102 omitted here) ,101,111,121,131,141,151,161,171,181,191,202,212,222,232,242,252,262,272,282,
A007602 Zuckerman numbers; divisible by product of digits ,1,2,3,4,5,6,7,8,9,11,12,15,24,36,111,112,115,128,132,135,144,175,212,216,224,
A104233 positive integers with a compact representation using + − * / ^ () ,125,128,216,243,256,343,512,625,729,
A006968 number of letters in Roman numeral representation of n ,1,2,3,2,1,2,3,4,2,1,2,3,4,3,2,3,4,5,3,2,3,4,5,4,3,4,5,6,4,3,4,5,6,5,4,5,6,7,
A118121 least number of Roman numerals needed to build n using + * () ,1,2,3,2,1,2,3,4,2,1,2,3,4,3,2,3,4,4,3,2,3,4,5,4,3,4,5,5,4,3,4,5,5,5,4,4,5,5,

Other bases

All sequences are shown in decimal.

OEIS Description First terms
A376897 digits, all are distinct in a2 in base 8 ,1,2,4,5,7,13,14,15,18,20,21,28,30,37,39,43,44,45,53,55,63,78,84,103,110,113,
A376898 digits, all are distinct in a3 in base 8 ,1,2,5,7,10,11,14,15,22,30,37,41,49,61,74,98,122,
A080790 emirps (see above) in base 2 ,11,13,23,29,37,41,43,47,53,61,67,71,83,97,101,113,131,151,163,167,173,181,
A049445 Harshad/Niven numbers (see above) in base 2 ,1,2,4,6,8,10,12,16,18,20,21,24,32,34,36,40,42,48,55,60,64,66,68,69,72,80,81,
A064150 Harshad/Niven numbers in base 3 ,1,2,3,4,6,8,9,10,12,15,16,18,20,21,24,25,27,28,30,32,33,35,36,39,40,45,48,54,
A064438 Harshad/Niven numbers in base 4 ,1,2,3,4,6,8,9,12,16,18,20,21,24,28,30,32,33,35,36,40,42,48,50,52,54,60,63,64,
A064481 Harshad/Niven numbers in base 5 ,1,2,3,4,5,6,8,10,12,15,16,18,20,24,25,26,27,28,30,32,36,40,42,45,48,50,51,52,
A245802 Harshad/Niven numbers in base 8 ,1,2,3,4,5,6,7,8,14,16,21,24,28,32,35,40,42,48,49,56,64,66,70,72,75,77,84,88,
A241989 Harshad/Niven numbers in base 16 ,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,18,20,30,32,33,35,36,40,45,48,50,54,
A006995 palindromes in base 2 (terms < 22 omitted here) ,5,7,9,15,17,21,27,31,33,45,51,63,65,73,85,93,99,107,119,127,129,153,165,189,
A014190 palindromes in base 3 (terms < 32 omitted here) ,10,13,16,20,23,26,28,40,52,56,68,80,82,91,100,112,121,130,142,151,160,164,
A014192 palindromes in base 4 (terms < 42 omitted here) ,17,21,25,29,34,38,42,46,51,55,59,63,65,85,105,125,130,150,170,190,195,215,
A029952 palindromes in base 5 (terms < 52 omitted here) ,26,31,36,41,46,52,57,62,67,72,78,83,88,93,98,104,109,114,119,124,126,156,186,
A029953 palindromes in base 6 (terms < 62 omitted here) ,37,43,49,55,61,67,74,80,86,92,98,104,111,117,123,129,135,141,148,154,160,166,
A029954 palindromes in base 7 (terms < 72 omitted here) ,50,57,64,71,78,85,92,100,107,114,121,128,135,142,150,157,164,171,178,185,192,
A029803 palindromes in base 8 (terms < 82 omitted here) ,65,73,81,89,97,105,113,121,130,138,146,154,162,170,178,186,195,203,211,219,227,
A029955 palindromes in base 9 (terms < 92 omitted here) ,82,91,100,109,118,127,136,145,154,164,173,182,191,200,209,218,227,236,246,
A029956 palindromes in base 11 (terms < 112 omitted here) ,122,133,144,155,166,177,188,199,210,221,232,244,255,266,277,288,299,310,321,
A029957 palindromes in base 12 (terms < 122 omitted here) ,145,157,169,181,193,205,217,229,241,253,265,277,290,302,314,326,338,350,362,
A029958 palindromes in base 13 (terms < 132 omitted here) ,170,183,196,209,222,235,248,261,274,287,300,313,326,340,353,366,379,392,405,
A029959 palindromes in base 14 (terms < 142 omitted here) ,197,211,225,239,253,267,281,295,309,323,337,351,365,379,394,408,422,436,450,
A029960 palindromes in base 15 (terms < 152 omitted here) ,226,241,256,271,286,301,316,331,346,361,376,391,406,421,436,452,467,482,497,
A029730 palindromes in base 16 (terms < 162 omitted here) ,257,273,289,305,321,337,353,369,385,401,417,433,449,465,481,497,514,530,546,
A075238 primes whose reversal in base 8 is also prime (incl. palindromes) ,2,3,5,7,13,29,31,41,43,47,59,61,67,71,73,79,89,97,101,107,113,193,211,227,
A278909 Smith numbers (see above) in base 2 ,15,51,55,85,125,159,185,190,205,215,222,238,246,249,253,287,303,319,374,407,

See also