login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 29 16:58 EST 2020. Contains 331347 sequences. (Running on oeis4.)