A258706 Absolute primes: every permutation of digits is a prime. Only the smallest representative of each permutation class is shown.

2, 3, 5, 7, 11, 13, 17, 37, 79, 113, 199, 337, 1111111111111111111, 11111111111111111111111

Offset: 1

Author

N. J. A. Sloane, Jun 09 2015

Keywords

• nonn
• base

Comments

See the main entry, A003459, for further information and references cited below.

The next terms are the repunit primes (A004023) R(317), too large to be displayed here, and R(1031), too large even for a b-file. Johnson (1977) proves that subsequent terms must be of the form a*R(n) + b*10^k, with a and a+b in {1..9}, k < n, and n > 9*10^9 if b != 0. - M. F. Hasler, Jun 26 2018

Links

• M. F. Hasler, Table of n, a(n) for n = 1..15
• James Grime and Brady Haran, Absolute Primes, YouTube Numberphile video, 2024.

Crossrefs

Cf. A003459, A004023, A004022 (subsequence of repunit primes).

Programs

Haskell

import Data.List (permutations, (\\))
a258706 n = a258706_list !! (n-1)
a258706_list = f a000040_list where
   f ps'@(p:ps) | any (== 0) (map a010051' dps) = f ps
                | otherwise = p : f (ps' \\ dps)
                where dps = map read $ permutations $ show p
-- Reinhard Zumkeller, Jun 10 2015

Mathematica

Flatten@{2, 3, 5, 7,
  Table[Select[
    Table @@
      Prepend[Prepend[
        Table[{A@k, A[k - 1], 4}, {k, 2, n}], {A[1], 4}],
       Unevaluated[
        Unevaluated[FromDigits[{1, 3, 7, 9}[[A /@ Range[n]]]]]]] //
     Flatten,
    Function[L,
       And[PrimeQ[#],
        And @@ PrimeQ[
          FromDigits /@ (Permute[L, #] & /@
             RandomPermutation[Length@L, 5])],
        And @@ PrimeQ[FromDigits /@ Rest[Permutations[L]]]]]@
      IntegerDigits@# &], {n, 2, 33}]}
(* Exhaustively searches thru 33 digits in ~7.5 sec, and up to 69 digits in 5 min, but cannot reach 317 digits. Not helpful in the light of Schroeppel's theorem that it's all repunits past 991. - Bill Gosper, Jan 06 2017 *)

PARI

{A=[2,5]; for(n=1, 317, my(D=[1,3,7,9], r=10^n\9); for(a=1,4, for(b=a^(n<3),4, for(j=0, if(b!=a,n-1), ispseudoprime(D[a]*r+(D[b]-D[a])*10^j)||next(2)); A=setunion(A, [r*D[a]+(D[b]-D[a])*10^if(b=n[1] && #select(d->d,n[^1]-n[^-1])<2 && !for(i=1,(#n)^(n[#n]>1), isprime(fromdigits(n=concat(n[^1],n[1])))||return)} \\ By Johnson's theorem and minimality required here, the number must be of the form ab...b or a...ab (=> first difference of digits has at most 1 nonzero component) and then is sufficient to consider rotations of the digits.
\\ M. F. Hasler, Jun 26 2018