A090987 -id:A090987 - cp's OEIS frontend

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

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

Hans Havermann, Feb 28 2004

nonn

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

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

Alexander Adamchuk, Aug 28 2006, Table of n, a(n) for n = 1..152 (term 3886 added by Daniel Suteu)
The Bernoulli Number Page, Table of factors of the numerators of Bernoulli numbers Bn in the range n = 2..10000, 2018.
S. S. Wagstaff, Jr, Prime factors of the absolute values of Bernoulli numerators, 2018.

Cf. A000367, A090943, A094095. For the smallest square factor, see A090987.

Cf. A122270, A122271, A122272, A122273.

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

A122270 Numbers m such that the numerator of the Bernoulli number B(m) is divisible by a cube.

Original entry on oeis.org

250, 686, 750, 1250, 1372, 1750, 2250, 2662, 2744, 2750, 3250, 3430, 3750, 4250, 4394, 4750, 4802, 5250, 5488, 5750, 6250, 6750, 6860, 7250, 7546, 7750, 7986, 8250, 8750, 8788, 8918, 9250, 9604, 9750, 9826

Offset: 1

Alexander Adamchuk, Aug 28 2006

nonn

For each m in the current sequence, the smallest prime whose cube divides the numerator of the Bernoulli number B(m) is listed in A122271.
The current sequence is a subset of A090997, which are numbers m such that the numerator of the Bernoulli number B(m) is divisible by a square.
A subset of the current sequence is A122272, which are numbers m such that the numerator of the Bernoulli number B(m) is divisible by a fourth power.
Conjecture: For all regular primes p > 3 and integers k > 0, the numerator of the Bernoulli number B(2*p^k) is divisible by p^k. Moreover, for all regular primes p > 3 and integers k > 0, m = 2*p^k is the smallest index such that the numerator of the Bernoulli number B(m) is divisible by p^k. Also, for all regular primes p > 3 and integers k > 0, all m such that the numerator of the Bernoulli number B(m) is divisible by p^k are of the form m = 2*s*p^k, where s > 0 is an integer.

			a(1) = 250 because it is the smallest number m such that numerator(B(m)) == 0 (mod 5^3). Note that 250 = 2*5^3.
a(2) = 686 because it is the smallest number m such that numerator(B(m)) == 0 (mod 7^3). Note that 686 = 2*7^3.

The Bernoulli Number Page, Table of factors of the numerators of Bernoulli numbers Bn in the range n = 2..10000, 2018.
S. S. Wagstaff, Jr, Prime factors of the absolute values of Bernoulli numerators, 2018.

Cf. A000367, A090987, A090997, A122271, A122272, A122273.

Various sections edited by Petros Hadjicostas, May 12 2020

A122271 a(n) is the smallest prime whose cube divides the numerator of the Bernoulli number B(A122270(n)).

Original entry on oeis.org

5, 7, 5, 5, 7, 5, 5, 11, 7, 5, 5, 7, 5, 5, 13, 5, 7, 5, 7, 5, 5, 5, 7, 5, 7, 5, 11, 5, 5, 13, 7, 5, 7, 5, 17

Offset: 1

Alexander Adamchuk, Aug 28 2006

			a(1) = 5 because 5^3 divides numerator(B(A122270(1))) = numerator(B(250)), but 3^3 does not.

The Bernoulli Number Page, Table of factors of the numerators of Bernoulli numbers Bn in the range n = 2..10000, 2018.
S. S. Wagstaff, Jr, Prime factors of the absolute values of Bernoulli numerators, 2018.

Cf. A000367, A090997, A090987, A122270, A122272, A122273.

Various sections edited by Petros Hadjicostas, May 12 2020

A122272 Numbers m such that the numerator of the Bernoulli number B(m) is divisible by p^4, where p is prime.

Original entry on oeis.org

1250, 3750, 4802, 6250, 8750, 9604

Offset: 1

Alexander Adamchuk, Aug 28 2006

For each m in the current sequence, the smallest prime p such that p^4 divides the numerator of the Bernoulli number B(m) is listed in A122273.
Note that the numerator of B(6250) is divisible by 5^5.
The current sequence is a subsequence of A122270, which are the numbers m such that the numerator of the Bernoulli number B(m) is divisible by a cube.
Sequence A122270 itself is a subsequence of A090997, which are the numbers m such that the numerator of the Bernoulli number B(m) is divisible by a square. [Edited by Petros Hadjicostas, May 12 2020]

			a(1) = 1250 because 5^4 divides numerator(B(1250)).
a(3) = 4802 because 7^4 divides numerator(B(4802)).

The Bernoulli Number Page, Table of factors of the numerators of Bernoulli numbers Bn in the range n = 2..10000, 2018.
S. S. Wagstaff, Jr, Prime factors of the absolute values of Bernoulli numerators, 2018.

Cf. A000367, A090987, A090997, A122270, A122271, A122273.

A122273 a(n) is the smallest prime p such that p^4 divides the numerator of the Bernoulli number B(A122272(n)).

Original entry on oeis.org

5, 5, 7, 5, 5, 7

Offset: 1

Alexander Adamchuk, Aug 28 2006

The numbers m in A122272 are such that the numerator of the Bernoulli number B(m) is divisible by p^4, where p is a prime. For m = 6250, we have that the numerator of B(6250) is divisible by 5^5.

The Bernoulli Number Page, Table of factors of the numerators of Bernoulli numbers Bn in the range n = 2..10000, 2018.
S. S. Wagstaff, Jr, Prime factors of the absolute values of Bernoulli numerators, 2018.

Cf. A000367, A090987, A090997, A122270, A122271, A122272.

Various sections edited by Petros Hadjicostas, May 12 2020

A090088 Smallest even pseudoprimes to odd base=2n-1, not necessarily exceeding n. See also A007535 and A090086, A090087.

Original entry on oeis.org

4, 286, 4, 6, 4, 10, 4, 14, 4, 6, 4, 22, 4, 26, 4, 6, 4, 34, 4, 38, 4, 6, 4, 46, 4, 10, 4, 6, 4, 58, 4, 62, 4, 6, 4, 10, 4, 74, 4, 6, 4, 82, 4, 86, 4, 6, 4, 94, 4, 14, 4, 6, 4, 106, 4, 10, 4, 6, 4, 118, 4, 122, 4, 6, 4, 10, 4, 134, 4, 6, 4, 142, 4, 146, 4, 6, 4, 14, 4, 158, 4, 6, 4, 166, 4, 10

Offset: 1

Labos Elemer, Nov 25 2003

nonn

For an even base there are no even pseudoprimes.

			n=2, 2n-2=3 as base, smallest relevant power is -1+2^(286-1) which is divisible by 286.

Antti Karttunen, Table of n, a(n) for n = 1..16520
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A007535, A090986, A090987, A090089.

Array[Block[{k = 4}, While[PowerMod[2 # - 1, k - 1, k] != 1, k += 2]; k] &, 86] (* Michael De Vlieger, Nov 13 2018 *)

A090088(n) = { forstep(k=4, oo, 2, if(1==(Mod(n+n-1, k)^(k-1)), return (k)); ); } \\ (After code in A090086) - Antti Karttunen, Nov 10 2018

a(n) = Min_{x=even number; (-1 + n^(x-1)) mod x = 0}.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A122270 Numbers m such that the numerator of the Bernoulli number B(m) is divisible by a cube.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A122271 a(n) is the smallest prime whose cube divides the numerator of the Bernoulli number B(A122270(n)).

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

A122272 Numbers m such that the numerator of the Bernoulli number B(m) is divisible by p^4, where p is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A122273 a(n) is the smallest prime p such that p^4 divides the numerator of the Bernoulli number B(A122272(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A090088 Smallest even pseudoprimes to odd base=2n-1, not necessarily exceeding n. See also A007535 and A090086, A090087.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula