A342320 Integers k such that Euler(k, 1) is an integer multiple of Bernoulli(k + 1, 1).

0, 1, 5, 17, 41, 53, 125, 161, 293, 341, 377, 485, 881, 1025, 1133, 1313, 1457, 1805, 2057, 2393, 2645, 3077, 3401, 3941, 4373, 5333, 5417, 6173, 6497, 7181, 7937, 9197, 9233, 10205, 11825, 12641, 13121, 14153, 14405, 16001, 16253, 16757, 18521, 19493, 21545

Offset: 0

Author

Peter Luschny, Mar 24 2021

Keywords

• nonn

Examples

			Let E(n) = Euler(n, 1) and B(n) = Bernoulli(n, 1).
2*E(0)  = 4*B(1) = 2;
2*E(1)  = 6*B(2) = 1;
2*E(5)  = 42*B(6) = 1;
2*E(17) = 58254*B(18) = 3202291;
2*E(41) = 418861572486*B(42) = 352552873457246307069012458671.
		

Crossrefs

a(n) = A015942(n-1)-1 for n>=2, (a(n)+1)/2 = A014945(n) for n>=1.

a(n) = A014741(n+1) - 1. - Vaclav Kotesovec, Mar 24 2021

Cf. A341759 (subsequence of primes), A198631/A006519 (Euler), A164555/A027642 (Bernoulli).

Programs

Mathematica

Join[{0}, Select[Range[1000], BernoulliB[#+1, 1] != 0 && IntegerQ[EulerE[#, 1]/BernoulliB[#+1, 1]] &]] (* Vaclav Kotesovec, Mar 24 2021 *)
Select[Range[100000], IntegerQ[(2*(-1 + 2^#))/#] & ] - 1 (* Vaclav Kotesovec, Mar 24 2021 *)
L342320 := Select[Range[0, 10000], Divisible[2^(# + 2) - 2, # + 1] &];
A342320[n_] := L342320[[n + 1]] (* Peter Luschny, Apr 10 2021 *)

Formula

Numbers k such that k + 1 divides 2^(k + 2) - 2. - Vaclav Kotesovec, Mar 24 2021