A220814 The elements of the set P5 in ascending order.

2, 3, 7, 13, 17, 19, 29, 37, 43, 53, 59, 73, 79, 97, 103, 107, 109, 113, 127, 137, 149, 157, 163, 173, 193, 197, 223, 227, 229, 233, 239, 257, 293, 307, 313, 317, 337, 347, 349, 379, 389, 409, 433, 439, 443, 449, 457, 467, 479, 487

Offset

1,1

Comments

P5 is the largest set of primes satisfying the conditions: (1) 5 is not in P5; (2) a prime p>5 is in P5 iff all prime divisors of p-1 are in P5.

P5 is also the set of all primes p for which the Pratt tree for p has no node labeled 5.

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

7 is in P5, because 7-1 = 2*3 and 2, 3 are in P5.

Mathematica

P5 = {2}; For[p = 2, p < 1000, p = NextPrime[p], If[p != 5 && AllTrue[ FactorInteger[p - 1][[All, 1]], MemberQ[P5, #] &], AppendTo[P5, p]]];
P5 (* 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(5, n) \\ Charles R Greathouse IV, Dec 26 2012

Crossrefs

Cf. A220813, A220815.

Sequence in context: A045327 A181103 A329487 * A045328 A045329 A271666

Adjacent sequences: A220811 A220812 A220813 * A220815 A220816 A220817

Keyword

nonn

Author

Franz Vrabec, Dec 22 2012

Status

approved