cp's OEIS Frontend

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

%I A122270 #26 May 13 2020 08:52:47
%S A122270 250,686,750,1250,1372,1750,2250,2662,2744,2750,3250,3430,3750,4250,
%T A122270 4394,4750,4802,5250,5488,5750,6250,6750,6860,7250,7546,7750,7986,
%U A122270 8250,8750,8788,8918,9250,9604,9750,9826
%N A122270 Numbers m such that the numerator of the Bernoulli number B(m) is divisible by a cube.
%C A122270 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.
%C A122270 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.
%C A122270 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.
%C A122270 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.
%H A122270 The Bernoulli Number Page, <a href="https://www.bernoulli.org/download/bn_factors.txt">Table of factors of the numerators of Bernoulli numbers Bn in the range n = 2..10000</a>, 2018.
%H A122270 S. S. Wagstaff, Jr, <a href="http://www.cerias.purdue.edu/homes/ssw/bernoulli/bnum">Prime factors of the absolute values of Bernoulli numerators</a>, 2018.
%e A122270 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.
%e A122270 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.
%Y A122270 Cf. A000367, A090987, A090997, A122271, A122272, A122273.
%K A122270 nonn
%O A122270 1,1
%A A122270 _Alexander Adamchuk_, Aug 28 2006
%E A122270 Various sections edited by _Petros Hadjicostas_, May 12 2020