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The elements of the set P7 in ascending order.
3

%I #21 Jan 05 2019 12:44:16

%S 2,3,5,11,13,17,19,23,31,37,41,47,53,61,67,73,79,83,89,97,101,103,107,

%T 109,131,137,139,149,151,157,163,167,179,181,191,193,199,223,229,241,

%U 251,257,263,269,271,277,283,293,307,311

%N The elements of the set P7 in ascending order.

%C 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.

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

%C It is conjectured that this sequence is infinite.

%H Ivan Neretin, <a href="/A220815/b220815.txt">Table of n, a(n) for n = 1..10000</a>

%H K. Ford, S. Konyagin and F. Luca, <a href="http://arxiv.org/abs/0904.0473">Prime chains and Pratt trees</a>, arXiv:0904.0473 [math.NT], 2009-2010; Geom. Funct. Anal., 20 (2010), pp. 1231-1258.

%H Kevin Ford, <a href="http://arxiv.org/abs/1212.3498">Sieving by very thin sets of primes, and Pratt trees with missing primes</a>, arXiv preprint arXiv:1212.3498 [math.NT], 2012-2013.

%F Ford proves that a(n) >> n^k for some k > 1. - _Charles R Greathouse IV_, Dec 26 2012

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

%t P7 = {2}; For[p = 2, p < 1000, p = NextPrime[p], If[p != 7 && AllTrue[ FactorInteger[p - 1][[All, 1]], MemberQ[P7, #] &], AppendTo[P7, p]]];

%t P7 (* _Jean-François Alcover_, Jan 05 2019 *)

%o (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)

%o is(n)=isprime(n) && P(7,n) \\ _Charles R Greathouse IV_, Dec 26 2012

%Y Cf. A220813, A220814.

%K nonn

%O 1,1

%A _Franz Vrabec_, Dec 22 2012