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A003459 Absolute primes: every permutation of 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."

This sequence has no terms with 4, 5 and 6 digits (by exhaustive search). - Sébastien Dumortier, Jun 16 2005

Depending on the source, permutable or absolute primes are sometimes required to have at least two different digits. This produces the subsequence A129338. - M. F. Hasler, Mar 26 2008


Angell, I. O. and Godwin, H. J. "On Truncatable Primes." Math. Comput. 31, 265-267, 1977.

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 (N0. 1, 1982), 38-41. [From N. J. A. Sloane, Jan 19 2012]

A. W. Johnson, Absolute primes, Mathematics Magazine, 1977, vol. 50, pp. 100-103.

Rich Schroeppel , personal communication.

Waclaw 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.

C. Caldwell, The prime glossary: Permutable Prime

J. P. Delahaye, Persistent Primes, Illustrating Permutable, Circular, Right & Left Truncatable Primes [broken link]

R. Ondrejka, The Top Ten: a Catalogue of Primal Configurations

W. Schneider, MATHEWS, Circular, Permutable, Truncatable and Deletable Primes

A. Slinko, Absolute Primes

Wikipedia, Permutable prime

Index entries for sequences related to truncatable primes


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*)



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


Includes all of A004022 = A002275(A004023).

Cf. A129338.

Cf. A010051.

Sequence in context: A234901 A090934 A068652 * A202264 A253717 A186307

Adjacent sequences:  A003456 A003457 A003458 * A003460 A003461 A003462




N. J. A. Sloane


The next terms are R(317), R(1031), R(49081), where R(n) is (10^n-1)/9.

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



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Last modified May 29 16:37 EDT 2015. Contains 257939 sequences.