cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A094095 a(n)^2 is the smallest square associated with the n-th term of A090943.

Original entry on oeis.org

103, 37, 59, 271, 37, 37, 67, 37, 59, 37, 101, 157, 37, 67, 59, 37, 37
Offset: 1

Views

Author

N. J. A. Sloane, May 02 2004

Keywords

Examples

			103^2 divides the numerator of the Bernoulli number B(228) (a 265-digit number).

Crossrefs

Cf. A090943, A092681.

Extensions

a(4)-a(17) from T. D. Noe, May 03 2004
Name clarified by Petros Hadjicostas, May 12 2020

A092224 Numbers k such that the numerator of Bernoulli(2*k) is divisible by 103, the fifth irregular prime.

Original entry on oeis.org

12, 63, 103, 114, 165, 206, 216, 267, 309, 318, 369, 412, 420, 471, 515, 522, 573, 618, 624, 675, 721, 726, 777, 824, 828, 879, 927, 930, 981, 1030, 1032, 1083, 1133, 1134, 1185, 1236, 1287, 1338, 1339, 1389, 1440, 1442, 1491, 1542, 1545, 1593, 1644, 1648
Offset: 1

Views

Author

Robert G. Wilson v, Feb 25 2004

Keywords

Comments

103 = A094095(1) is the first irregular prime in A094095. This sequence is the union of 2 arithmetic progressions: (24 + 102*n)/2 and 103*n. Note that the numerator of BernoulliB(2*114) is divisible by the first nontrivial irregular squared prime 103^2, when A090943(1)/2 = a(n) = 114 = (24 + 102*2)/2. Also, the numerator of BernoulliB(2*1236) is divisible by 103^2 because a(n) = 1236 = (24 + 102*24)/2 = 103*24/2. - Alexander Adamchuk, Jul 31 2006

Links

Crossrefs

Cf. A000928, A091216, A092221, A092222, A092223, A092225, A092226, A092227, A092228, A092229.
Cf. A094095, A090943, A027641.

Programs

  • Mathematica
    Select[ Range[ 1694], Mod[ Numerator[ BernoulliB[2# ]], 103] == 0 &]
Select[Union[Table[2n*103,{n,1,100}],Table[24+102*n,{n,0,100}]], #<=10000&]/2 (* Alexander Adamchuk, Jul 31 2006 *)

A090997 Numbers m such that the numerator of the Bernoulli number B(m) is divisible by a square.

Original entry on oeis.org

50, 98, 150, 196, 228, 242, 250, 284, 338, 350, 392, 450, 484, 490, 550, 578, 650, 676, 686, 722, 726, 750, 784, 850, 914, 950, 968, 980, 1014, 1050, 1058, 1078, 1150, 1156, 1184, 1250, 1274, 1350, 1352, 1372, 1434, 1444, 1450, 1452, 1550, 1568, 1616
Offset: 1

Views

Author

Hans Havermann, Feb 28 2004

Keywords

Comments

It appears that all terms that are divisible by p^2 and do not belong to A090943 are of the form 2*k*p^2, where p is a prime and k > 0 is an integer. Also, all numbers in A090943 are terms because they are divisible by the squares of irregular primes in A094095. The corresponding smallest primes p such that their squares divide terms are listed in A090987. - Alexander Adamchuk, Aug 19 2006
A subsequence of the current sequence is A122270, which are the numbers m such that the numerator of the Bernoulli number B(m) is divisible by a cube. Another subsequence of the current sequence is A122272, which are the numbers m such that the numerator of the Bernoulli number B(m) is divisible by p^4, where p is a prime. Note that the numerator of the Bernoulli number B(6250) is divisible by 5^5. - Alexander Adamchuk, Aug 28 2006

Examples

			a(3) = 150 because numerator(B(150)) == 0 (mod 5^2).

Links

Crossrefs

Cf. A000367, A090943, A094095. For the smallest square factor, see A090987.
Cf. A122270, A122271, A122272, A122273.

Extensions

In view of the phrase "it appears", it is not clear to me that the correctness of this sequence has been rigorously established. - N. J. A. Sloane, Aug 26 2006
More terms from Alexander Adamchuk, Aug 19 2006
More terms from Alexander Adamchuk, Aug 28 2006
Various sections edited by Petros Hadjicostas, May 12 2020
Incorrect term 294 removed by Daniel Suteu, May 21 2020
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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