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

%I A342320 #23 Apr 10 2021 08:08:59
%S A342320 0,1,5,17,41,53,125,161,293,341,377,485,881,1025,1133,1313,1457,1805,
%T A342320 2057,2393,2645,3077,3401,3941,4373,5333,5417,6173,6497,7181,7937,
%U A342320 9197,9233,10205,11825,12641,13121,14153,14405,16001,16253,16757,18521,19493,21545
%N A342320 Integers k such that Euler(k, 1) is an integer multiple of Bernoulli(k + 1, 1).
%F A342320 Numbers k such that k + 1 divides 2^(k + 2) - 2. - _Vaclav Kotesovec_, Mar 24 2021
%e A342320 Let E(n) = Euler(n, 1) and B(n) = Bernoulli(n, 1).
%e A342320 2*E(0)  = 4*B(1) = 2;
%e A342320 2*E(1)  = 6*B(2) = 1;
%e A342320 2*E(5)  = 42*B(6) = 1;
%e A342320 2*E(17) = 58254*B(18) = 3202291;
%e A342320 2*E(41) = 418861572486*B(42) = 352552873457246307069012458671.
%t A342320 Join[{0}, Select[Range[1000], BernoulliB[#+1, 1] != 0 && IntegerQ[EulerE[#, 1]/BernoulliB[#+1, 1]] &]] (* _Vaclav Kotesovec_, Mar 24 2021 *)
%t A342320 Select[Range[100000], IntegerQ[(2*(-1 + 2^#))/#] & ] - 1 (* _Vaclav Kotesovec_, Mar 24 2021 *)
%t A342320 L342320 := Select[Range[0, 10000], Divisible[2^(# + 2) - 2, # + 1] &];
%t A342320 A342320[n_] := L342320[[n + 1]] (* _Peter Luschny_, Apr 10 2021 *)
%Y A342320 a(n) = A015942(n-1)-1 for n>=2, (a(n)+1)/2 = A014945(n) for n>=1.
%Y A342320 a(n) = A014741(n+1) - 1. - _Vaclav Kotesovec_, Mar 24 2021
%Y A342320 Cf. A341759 (subsequence of primes), A198631/A006519 (Euler), A164555/A027642 (Bernoulli).
%K A342320 nonn
%O A342320 0,3
%A A342320 _Peter Luschny_, Mar 24 2021