cp's OEIS Frontend

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

It appears that, except for irregular primes belonging to A094095, such as a(5) = 103, a(8) = 37 and a(26) = 59, all regular prime a(n) = p divide the corresponding numerators of the Bernoulli numbers B(A090997(n)) with indices of the form 2*k*p^2, where k > 0 is an integer. - Alexander Adamchuk, Aug 19 2006

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.

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]