%I #34 Feb 10 2021 20:54:37

%S 1,2,3,5,11,13,19,29,37,53,59,61,67,83,101,107,131,139,149,163,173,

%T 179,181,197,211,227,269,293,317,347,349,373,379,389,419,421,443,461,

%U 467,491,509,523,541,547,557,563,587,613,619,653,659,661,677,701,709,757,773,787,797,821,827,829,853,859,877,883,907,941,947

%N Numbers k such that for any x in 1..k-1 there exists a y in 0..k-2 such that x^2 == 2^y (mod k).

%C It seems that the sequence consists of {1,2} union A001122. The sequence differs from A082595 because here the multiplicity is not important (see example: P contains two 5's and Q is required to have at least one 5, not necessarily 2 5's).

%C Numbers k for which there is a permutation of 0..k-1 such that each number is the sum of all the previous numbers, plus 1, mod k. - _R. H. Hardin_, Dec 28 2007

%C Positive numbers k such that x^(k-1) + x^(k-2) + x^(k-3) + ... + x + 1 is irreducible over GF(2). - _Arkadiusz Wesolowski_, Nov 20 2012

%C All terms > 1 are prime. Appears to be set of numbers k such that the sequence 2^n mod k has period length of k-1. - _Gary Detlefs_, May 15 2014

%F a(n) = A071642(n) + 1. - _Arkadiusz Wesolowski_, Nov 20 2012

%e The set of values for x^2 mod 19, 1<=x<=18, is P=[1, 4, 9, 16, 6, 17, 11, 7, 5, 5, 7, 11, 17, 6, 16, 9, 4, 1], the set of values for 2^y mod 19, 0<=y<=n-2 is Q= [1, 2, 4, 8, 16, 13, 7, 14, 9, 18, 17, 15, 11, 3, 6, 12, 5, 10] which contains all values in P, hence 19 is in the sequence.

%o (PARI) for(n=1,450,if(sum(y=1,n-1,if(setsearch(Set(vector(n-1,x,2^(x-1)%n)),y),0,1))==0,print1(n,",")))

%Y Cf. A001122, A082595.

%K nonn

%O 1,2

%A _Benoit Cloitre_, May 08 2003

%E More terms from _R. H. Hardin_, Dec 28 2007