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 p-1 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. 1231-1258 (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], 2012-2013.
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 11-1 = 2*5 and 2, 5 are in P3.
MAPLE
with(numtheory):
P3:= proc(n) P3(n):= `if`(n<1, {}, P3(n-1) union {a(n)}) end:
a:= proc(n) option remember; local p;
if n<3 then [2, 5][n]
else p:=a(n-1);
do p:= nextprime(p);
if factorset(p-1) minus P3(n-1) = {} 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[ p-1][[All, 1]], MemberQ[P3, #]&], AppendTo[P3, p]]]; P3 (* Jean-François Alcover, Feb 24 2016 *)
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(3, n) \\ Charles R Greathouse IV, Dec 26 2012
CROSSREFS
KEYWORD
nonn
AUTHOR
Franz Vrabec, Dec 22 2012
STATUS
approved