

A220815


The elements of the set P7 in ascending order.


3



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
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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 p1 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], 20092010; Geom. Funct. Anal., 20 (2010), pp. 12311258.
Kevin Ford, Sieving by very thin sets of primes, and Pratt trees with missing primes, arXiv preprint arXiv:1212.3498 [math.NT], 20122013.


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 111 = 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 (* JeanFrançois Alcover, Jan 05 2019 *)


PROG

(PARI) P(k, n)=if(n<=k, n<k, my(f=factor(n1)[, 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



