1,1

The smallest cube prime factor of the numerator of BernoulliB[(a(n)] is given in A122271[n] = {5,7,5,5,7,5,5,11,7,5,5,7,5,5,13,5,7,5,...}. a(n) is a subset of A090997[n] = {50,98,150,196,228,242,250,284,294,338,350,392,450,484,490,550,578,650,676,686,...} Numbers n such that numerator of Bernoulli number B(n) is divisible by a square. A subset of a(n) is A122272[n] = {1250,3750,4802,6250,8750,9604,...} Numbers n such that numerator of Bernoulli number B(n) is divisible by Prime[k]^4. Conjecture : For all regular primes p>3 and integers k>0 the numerator of Bernoulli number B(2p^k) is divisible by p^k. In fact, for all regular primes p>3 and integers k>0, n = 2p^k is the smallest index such that numerator of Bernoulli number B(n) is divisible by p^k. Also, for all regular primes p>3 and integers k>0, all n such that numerator of Bernoulli number B(n) is divisible by p^k are of the form n = 2m*p^k, where m>0 is an integer.

Table of factors of the numerators of Bernoulli numbers Bn in the range n=2,...,10000. at www.bernoulli.org

a(1) = 250 because it is the smallest number n such that Mod[ Numerator[ BernoulliB[n] ], 5^3] = 0. Note that 250 = 2*5^3.

a(2) = 686 because it is the smallest number n such that Mod[ Numerator[ BernoulliB[n] ], 7^3] = 0. Note that 686 = 2*7^3.

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

Sequence in context: A069154 A045169 A045185 this_sequence A109121 A165281 A033449

Adjacent sequences: A122267 A122268 A122269 this_sequence A122271 A122272 A122273

nonn

Alexander Adamchuk (alex(AT)kolmogorov.com), Aug 28 2006