A107845 Transposable-digit primes: Primes such that if any single pair of adjacent digits is transposed the result is a prime.

2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 137, 179, 199, 277, 311, 331, 337, 373, 379, 397, 419, 491, 577, 613, 733, 811, 877, 911, 919, 977, 991, 1013, 1031, 1091, 1117, 1213, 1231, 1579, 1777, 1811, 1999, 2113, 2131, 2399, 2411, 2677, 2699, 2719

Offset: 1

Author

Rick L. Shepherd, May 24 2005

Keywords

• base
• nonn

Comments

A single transposition error (of decimal digits) when recording or communicating such a prime still results in a prime (possibly the same prime). A003459 and A004022 (its subsequence) are subsequences. A003459 is also a subsequence of A068652.

Examples

			137 is a term because it is prime and 173 and 317 are also prime.
173 is not a term because 713 is not prime (even though 173 and 137 are prime). {Hence none of 137,173,317,371,713,731 is a term of A003459.}
3119 is a term because it is prime and 1319 and 3191 are primes.
As 3119, 1193, 1931 and 9311 are all prime, 3119 is also a term of A068652.
Finally, although 1913 is also prime, neither 1139, 1391, 3911, 9113, nor 9131 is prime so 3119's twelve total permutations are not terms of A003459.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..5000 (first 3565 terms from Felix Fröhlich)

Crossrefs

Cf. A003459 (absolute primes), A004022 (repunit primes), A068652 (every cyclic permutation is prime).

Programs

Mathematica

swap[lst_List, i_Integer] := Block[{lsu = lst}, ReplacePart[ lsu, {i -> lsu[[i + 1]], i + 1 -> lsu[[i]]}]]; fQ[n_] := Block[{id = IntegerDigits@ n, l = Floor@ Log10@ n}, And @@ Table[ PrimeQ@ FromDigits@ swap[id, j], {j, l}] == True]; Select[ Prime@ Range@ 500, fQ] (* Robert G. Wilson v, Nov 29 2014 *)

PARI

eva(n) = subst(Pol(n), x, 10)
switchdigits(v, pos) = my(vt=v[pos]); v[pos]=v[pos+1]; v[pos+1]=vt; v
is(n) = my(d=digits(n)); for(k=1, #d-1, if(!ispseudoprime(eva(switchdigits(d, k))), return(0))); 1
forprime(p=1, , if(is(p), print1(p, ", "))) \\ Felix Fröhlich, Sep 21 2019

Extensions

Offset changed from 0 to 1 by Felix Fröhlich, Sep 21 2019