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A003459 Absolute primes (or permutable primes): every permutation of the digits is a prime.
(Formerly M0658)
2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991, 1111111111111111111, 11111111111111111111111 (list; graph; refs; listen; history; text; internal format)



"The prime repunits are examples of integers which are prime and remain prime after an arbitrary permutation of their decimal digits. Integers with this property are called either 'permutable primes' according to H.-E. Richert, who introduced them some 40 years ago, or 'absolute primes' according to T. N. Bhagava and P. H. Doyle and A. W. Johnson."

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


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).


Table of n, a(n) for n=1..24.

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.

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.


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


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 *)



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

(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


Includes all of A004022 = A002275(A004023).

A258706 gives minimal representatives of the permutation classes.

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

Cf. A039986.

Sequence in context: A234901 A090934 A068652 * A276132 A202264 A253717

Adjacent sequences:  A003456 A003457 A003458 * A003460 A003461 A003462




N. J. A. Sloane


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

Additional comments from Robert G. Wilson v, Aug 18 2000



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Last modified September 19 02:54 EDT 2020. Contains 337175 sequences. (Running on oeis4.)