A003459 Absolute primes (or permutable primes): every permutation of the digits is a prime.

2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991, 1111111111111111111, 11111111111111111111111

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• base
• nice
• hard

Comments

From Bill Gosper, Jan 24 2003, in a posting to the Math Fun Mailing List: (Start)

Recall Sloane's old request for more terms of A003459 = (2 3 5 7 11 13 17 31 37 71 73 79 97 113 131 199 311 337 373 733 919 991 ...) and Richard C. Schroeppel's astonishing observation that the next term is 1111111111111111111. Absent Rich's analysis, trying to extend this sequence makes a great set of beginner's programming exercises. We may restrict the search to combinations of the four digits 1,3,7,9, only look at starting numbers with nondecreasing digits, generate only unique digit combinations, and only as needed. (We get the target sequence afterward by generating and merging the various permutations, and fudging the initial 2,3,5,7.)

To my amazement the (uncompiled, Macsyma) program printed 11,13,...,199,337, and after about a minute, 1111111111111111111!

And after a few more minutes, (10^23-1)/9! (End)

Boal and Bevis say that Johnson (1977) proves that if there is a term > 1000 with exactly two distinct digits then it must have more than nine billion digits. - N. J. A. Sloane, Jun 06 2015

Some authors require permutable or absolute primes to have at least two different digits. This produces the subsequence A129338. - M. F. Hasler, Mar 26 2008

See A039986 for a related problem with more sophisticated (PARI) code (iteration over only inequivalent digit permutations). - M. F. Hasler, Jul 10 2018

References

• Richard C. Schroeppel, personal communication.
• Wacław Sierpiński, Co wiemy, a czego nie wiemy o liczbach pierwszych. Warsaw: PZWS, 1961, pp. 20-21.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 113.

Links

• I. O. Angell and H. J. Godwin, On Truncatable Primes, Math. Comput. 31, 265-267, 1977. [Related paper, but primarily concerned with A023107 and A103443. - _N. J. A. Sloane_, Jun 06 2015]
• T. N. Bhargava and P. H. Doyle, On the existence of absolute primes, Math. Mag., 47 (1974), 233.
• J. L. Boal and J. H. Bevis, Permutable primes. Math. Mag., 55 (No. 1, 1982), 38-41.
• C. Caldwell, The prime glossary: Permutable Prime.
• J. P. Delahaye, Persistent Primes, Illustrating Permutable, Circular, Right & Left Truncatable Primes, Pour La Science no 256.
• James Grime and Brady Haran, Absolute Primes, YouTube Numberphile video, 2024.
• A. W. Johnson, Absolute primes, Mathematics Magazine, 1977, vol. 50, pp. 100-103.
• R. Ondrejka, The Top Ten: a Catalogue of Primal Configurations.
• W. Schneider, MATHEWS, Circular, Permutable, Truncatable and Deletable Primes.
• A. Slinko, Absolute Primes Oct. 2000.
• A. Slinko, Absolute Primes, Oct. 2000 [Cached copy, permission requested].
• Wikipedia, Permutable prime.
• Index entries for sequences related to truncatable primes.

Crossrefs

Includes all of A004022 = A002275(A004023).

A258706 gives minimal representatives of the permutation classes.

Cf. A010051, A023107, A016114, A103443, A129338, A141263.

Cf. A039986.

Programs

Haskell

import Data.List (permutations)
a003459 n = a003459_list !! (n-1)
a003459_list = filter isAbsPrime a000040_list where
   isAbsPrime = all (== 1) . map (a010051 . read) . permutations . show
-- Reinhard Zumkeller, Sep 15 2011

Mathematica

f[n_]:=Module[{b=Permutations[IntegerDigits[n]],q=1},Do[If[!PrimeQ[c=FromDigits[b[[m]]]],q=0;Break[]],{m,Length[b]}];q];Select[Range[1000],f[#]>0&] (* Vladimir Joseph Stephan Orlovsky, Feb 03 2011 *)
(* Linear complexity: can't reach R(19). See A258706. - Bill Gosper, Jan 06 2017 *)

PARI

for(n=1, oo, my(S=[],r=10^n\9); for(a=1, 9^(n>1), for(b=if(n>2, 1-a), 9-a, for(j=0, if(b, n-1), ispseudoprime(a*r+b*10^j)||next(2)); S=concat(S,vector(if(b,n,1),k,a*r+10^(k-1)*b))));apply(t->printf(t","),Set(S))) \\ M. F. Hasler, Jun 26 2018

Formula

Conjecture: for n >= 23, a(n) = A004022(n-21). - Max Alekseyev, Oct 08 2018

Extensions

The next terms are a(25)=A002275(317), a(26)=A002275(1031), a(27)=A002275(49081).