A050229 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).

1, 2, 3, 5, 11, 13, 19, 29, 37, 53, 59, 61, 67, 83, 101, 107, 131, 139, 149, 163, 173, 179, 181, 197, 211, 227, 269, 293, 317, 347, 349, 373, 379, 389, 419, 421, 443, 461, 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

Offset: 1

Author

Benoit Cloitre, May 08 2003

Keywords

• nonn

Comments

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).

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

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

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

Examples

			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.
		

Crossrefs

Cf. A001122, A082595.

Programs

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,",")))

Formula

a(n) = A071642(n) + 1. - Arkadiusz Wesolowski, Nov 20 2012

Extensions

More terms from R. H. Hardin, Dec 28 2007