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
KEYWORD
nonn
AUTHOR
Franz Vrabec, Dec 22 2012
STATUS
approved