

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

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.


FORMULA



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]]];


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)


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



