A220815 The elements of the set P7 in ascending order.

2, 3, 5, 11, 13, 17, 19, 23, 31, 37, 41, 47, 53, 61, 67, 73, 79, 83, 89, 97, 101, 103, 107, 109, 131, 137, 139, 149, 151, 157, 163, 167, 179, 181, 191, 193, 199, 223, 229, 241, 251, 257, 263, 269, 271, 277, 283, 293, 307, 311

Offset

1,1

Comments

P7 is the largest set of primes satisfying the conditions: (1) 7 is not in P7; (2) a prime p>7 is in P7 iff all prime divisors of p-1 are in P7.

P7 is also the set of all primes p for which the Pratt tree for p has no node labeled 7.

It is conjectured that this sequence is infinite.

Links

Ivan Neretin, Table of n, a(n) for n = 1..10000

K. Ford, S. Konyagin and F. Luca, Prime chains and Pratt trees, arXiv:0904.0473 [math.NT], 2009-2010; Geom. Funct. Anal., 20 (2010), pp. 1231-1258.

Kevin Ford, Sieving by very thin sets of primes, and Pratt trees with missing primes, arXiv preprint arXiv:1212.3498 [math.NT], 2012-2013.

Formula

Ford proves that a(n) >> n^k for some k > 1. - Charles R Greathouse IV, Dec 26 2012

Example

11 is in P7, because 11-1 = 2*5 and 2, 5 are in P7.

Mathematica

P7 = {2}; For[p = 2, p < 1000, p = NextPrime[p], If[p != 7 && AllTrue[ FactorInteger[p - 1][[All, 1]], MemberQ[P7, #] &], AppendTo[P7, p]]];
P7 (* Jean-François Alcover, Jan 05 2019 *)

Prog

(PARI) P(k, n)=if(n<=k, n<k, my(f=factor(n-1)[, 1]); for(i=1, #f, if(!P(k, f[i]), return(0))); 1)
is(n)=isprime(n) && P(7, n) \\ Charles R Greathouse IV, Dec 26 2012

Crossrefs

Cf. A220813, A220814.

Sequence in context: A040062 A115653 A042997 * A171600 A126148 A264866

Adjacent sequences: A220812 A220813 A220814 * A220816 A220817 A220818

Keyword

nonn

Author

Franz Vrabec, Dec 22 2012

Status

approved