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

250, 686, 750, 1250, 1372, 1750, 2250, 2662, 2744, 2750, 3250, 3430, 3750, 4250, 4394, 4750, 4802, 5250, 5488, 5750, 6250, 6750, 6860, 7250, 7546, 7750, 7986, 8250, 8750, 8788, 8918, 9250, 9604, 9750, 9826

Offset: 1

Author

Alexander Adamchuk, Aug 28 2006

Keywords

• nonn

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.

Examples

			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.
		

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.

Crossrefs

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

Extensions

Various sections edited by Petros Hadjicostas, May 12 2020