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A220814 The elements of the set P5 in ascending order. 3
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified December 11 02:13 EST 2019. Contains 329910 sequences. (Running on oeis4.)