OFFSET
1,1
COMMENTS
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.
LINKS
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.
EXAMPLE
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.
CROSSREFS
KEYWORD
nonn
AUTHOR
Alexander Adamchuk, Aug 28 2006
EXTENSIONS
Various sections edited by Petros Hadjicostas, May 12 2020
STATUS
approved