A383043 Integers k such that d*2^k - 1 is prime for some proper divisor d of k.

2, 3, 4, 5, 6, 7, 10, 12, 13, 16, 17, 18, 19, 21, 28, 30, 31, 36, 42, 46, 54, 60, 61, 63, 75, 88, 89, 99, 102, 104, 106, 107, 108, 126, 127, 132, 133, 204, 214, 216, 225, 264, 270, 286, 304, 306, 324, 330, 342, 352, 390, 414, 420, 456, 462, 468

Offset: 1

Author

Juri-Stepan Gerasimov, Apr 19 2025

Keywords

• nonn

Examples

			81 is not in the sequence because 1*2^81 - 1, 3*2^81 - 1, 9*2^81 - 1 and 27*81 - 1 are composites where 1, 3, 9 and 27 are proper divisors d of k = 81, while 81*2^81 - 1 is prime where 81 is nonproper divisor d of k = 81.
		

Crossrefs

Supersequence of A000043. Subsequence of A382811.

Cf. A002234, A032741.

Programs

Magma

[k: k in [1..500] | not #[d: d in [1..k-1] | k mod d eq 0 and IsPrime(d*2^k-1)] eq 0];

Mathematica

s={};Do[d=Drop[Divisors[n],-1];If[ContainsAny[PrimeQ[d*2^n-1],{True}],AppendTo[s,n]],{n,468}];s (* James C. McMahon, May 01 2025 *)

PARI

isok(k) = fordiv(k, d, if ((dMichel Marcus, Apr 20 2025