A057719 Prime factors of numbers in A006521 (numbers k that divide 2^k + 1).

3, 19, 163, 571, 1459, 8803, 9137, 17497, 41113, 52489, 78787, 87211, 135433, 139483, 144667, 164617, 174763, 196579, 274081, 370009, 370387, 478243, 760267, 941489, 944803, 1041619, 1220347, 1236787, 1319323, 1465129, 1663579, 1994659

Offset: 1

Author

Ignacio Larrosa Cañestro, Oct 26 2000

Keywords

• nonn

Comments

A prime p is in this sequence iff all prime divisors of ord_p(2)/2 are in this sequence, where ord_p(2) is the order of 2 modulo p. - Max Alekseyev, Jul 30 2006

Examples

			2^171 + 1 == 0 (mod 171), 171 = 3^2*19, 2^13203+1 == 0 (mod 13203), 13203 = 3^4*163.
		

Links

• Joerg Arndt, Table of n, a(n) for n = 1..220 (terms up to 10^9, terms for n = 1..100 from T. D. Noe)
• Alexander Kalmynin, On Novák numbers, arXiv:1611.00417 [math.NT], 2016. See Chapter 4 p. 7 Novák primes.
• C. Smyth, The terms in Lucas Sequences divisible by their indices, JIS 13 (2010) #10.2.4.

Crossrefs

Cf. A006521, A066364.

Cf. A136474, A136473.

Programs

Mathematica

S = {2}; Reap[For[p = 3, p < 2 10^6, p = NextPrime[p], f = FactorInteger[ MultiplicativeOrder[2, p]]; If[f[[1, 1]] != 2 || f[[1, 2]] != 1, Continue[]]; f = f[[All, 1]]; If[Length[Intersection[S, f]] == Length[f], S = Union[S, {p}]; Print[p]; Sow[p]]]][[2, 1]] (* Jean-François Alcover, Nov 11 2018, from PARI *)

PARI

{ A057719() = local(S,f); S=Set([2]); forprime(p=3,10^7, f=factorint(znorder(Mod(2,p))); if(f[1,1]!=2||f[1,2]!=1,next); f=f[,1]; if(length(setintersect(S,Set(f)))==length(f), S=setunion(S,[p]); print1(p,", "))) }

Extensions

Edited by Max Alekseyev, Jul 30 2006