

A220813


The elements of the set P3 in ascending order.


3



2, 5, 11, 17, 23, 41, 47, 83, 89, 101, 137, 167, 179, 251, 257, 353, 359, 401, 461, 503, 641, 719, 809, 821, 881, 941, 1013, 1097, 1151, 1283, 1361, 1409, 1433, 1439, 1601, 1619, 1871, 2027, 2069, 2351, 2531, 2657, 2663, 2741, 2789, 2819, 2879, 3203, 3209, 3581
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OFFSET

1,1


COMMENTS

P3 is the largest set of primes satisfying the conditions: (1) 3 is not in P3; (2) a prime p>3 is in P3 iff all prime divisors of p1 are in P3.
P3 is also the set of all primes p for which the Pratt tree for p has no node labeled 3.
It is conjectured that this sequence is infinite.


LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..10000
K. Ford, S. Konyagin and F. Luca, Prime chains and Pratt trees, Geom. Funct. Anal., 20 (2010), pp. 12311258 (arXiv:0904.0473 [math.NT]).
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. "It appears" that k can be taken as 1.612.  Charles R Greathouse IV, Dec 26 2012


EXAMPLE

11 is in P3, because 111 = 2*5 and 2, 5 are in P3.


MAPLE

with(numtheory):
P3:= proc(n) P3(n):= `if`(n<1, {}, P3(n1) union {a(n)}) end:
a:= proc(n) option remember; local p;
if n<3 then [2, 5][n]
else p:=a(n1);
do p:= nextprime(p);
if factorset(p1) minus P3(n1) = {} then break fi
od; p
fi
end:
seq(a(n), n=1..70); # Alois P. Heinz, Dec 26 2012


MATHEMATICA

P3 = {2, 5}; For[p=11, p<4000, p=NextPrime[p], If[ AllTrue[ FactorInteger[ p1][[All, 1]], MemberQ[P3, #]&], AppendTo[P3, p]]]; P3 (* JeanFrançois Alcover, Feb 24 2016 *)


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(3, n) \\ Charles R Greathouse IV, Dec 26 2012


CROSSREFS

Cf. A220814, A220815.
Sequence in context: A113426 A078894 A086319 * A217303 A053033 A136244
Adjacent sequences: A220810 A220811 A220812 * A220814 A220815 A220816


KEYWORD

nonn


AUTHOR

Franz Vrabec, Dec 22 2012


STATUS

approved



