A220813 The elements of the set P3 in ascending order.

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

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

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