A161790 The positive integer n is included if 1 is the largest integer of the form {2^k - 1} to divide n.

1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20, 22, 23, 25, 26, 29, 32, 34, 37, 38, 40, 41, 43, 44, 46, 47, 50, 52, 53, 55, 58, 59, 61, 64, 65, 67, 68, 71, 73, 74, 76, 79, 80, 82, 83, 85, 86, 88, 89, 92, 94, 95, 97, 100, 101, 103, 104, 106, 107, 109, 110, 113, 115, 116, 118, 121, 122, 125, 128, 130, 131, 134, 136, 137, 139, 142, 143

Offset: 1

Author

Leroy Quet, Jun 19 2009

Keywords

• nonn

Comments

Numbers which are not multiple of 2^k-1, k > 1. Because 2^k-1 = 1+2+...+2^(k-1), these numbers are also not the sum of positive integers in a geometric progression with common ratio 2 (cf. the primes A000040 which satisfy a similar property with arithmetic progressions with common difference 2). - Jean-Christophe Hervé, Jun 19 2014

The asymptotic density of this sequence is 1 - Sum_{s subset of A000225 \ {0, 1}} (-1)^(card(s)+1)/LCM(s) = 0.54830... - Amiram Eldar, Jun 30 2025

Links

• Diana L. Mecum, Table of n, a(n) for n = 1..2000

Crossrefs

Cf. A000225, A382875 (complement).

Positions of ones in A154402, A161788 and A161789.

Programs

Mathematica

DivisorList=Drop[Table[2^k-1,{k,1,20}],1]
A161790=Union[Table[If[Length[Join[DivisorList,Drop[Divisors[n],1]]]==Length[Union[DivisorList,Drop[Divisors[n],1]]],n,],{n,1,5000}]]
(* Second program: *)
Position[Table[DivisorSum[n, 1 &, IntegerQ@ Log2[# + 1] &], {n, 143}], 1][[All, 1]] (* Michael De Vlieger, Jun 11 2018 *)

PARI

isok(k) = forstep(i = logint(k+1, 2), 2, -1, if(k % (2^i-1) == 0, return(0))); 1; \\ Amiram Eldar, Jun 30 2025

Formula

A161788(a(n)) = A161789(a(n)) = 1.

Also A154402(a(n)) = 1. - Antti Karttunen, Jun 11 2018