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

Author

Hans Havermann, Feb 28 2004

Keywords

• nonn

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

• 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.

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