login
Prime numbers n such that replacing each digit d in the decimal expansion of n with d^d produces a prime. Zeros are not allowed.
2

%I #15 Sep 13 2017 00:05:56

%S 11,13,17,31,53,61,71,79,151,167,229,233,251,263,311,313,331,337,349,

%T 367,389,419,443,673,751,947,971,991,1433,1531,1699,1733,1993,2111,

%U 2141,2153,2221,2333,2393,2521,2833,2963,3137,3167,3323,3343,3371,3389,3391

%N Prime numbers n such that replacing each digit d in the decimal expansion of n with d^d produces a prime. Zeros are not allowed.

%C Subsequence of A240623.

%C If zeros are counted with the convention 0^0 = 1, we find the additional primes 409, 2011, 2027, 2053, 2063, 2081, 2503, 3037, 3061, 3067, 4093, 6029, 6079, 6203, 7001, 8011, 8101, 8807, 9043, 9403, 10103, 10141, 10211, 10321, 10513, 10663, 11003, 11027, 11503, 12037,...

%H Vincenzo Librandi, <a href="/A240624/b240624.txt">Table of n, a(n) for n = 1..1000</a>

%e 263 is in the sequence because 263 becomes 44665627 which is also prime, where 44665627 is the concatenation (2^2, 6^6, 3^3) = (4, 46656, 27).

%t lst={};f[n_]:=Block[{a=IntegerDigits[n],b="",k=1,l},l=Length[a];While[k<l+1,b=StringJoin[b,ToString[a[[k]]^a[[k]]]];k++];ToExpression[b]];Do[If[PrimeQ[f[Prime[n]]],AppendTo[lst,Prime[n]]],{n,1,600}];lst

%t ddQ[n_]:=Module[{idn=IntegerDigits[n]},!MemberQ[idn,0]&&PrimeQ[FromDigits[ Flatten[ IntegerDigits/@ (idn^idn)]]]]; Select[Prime[Range[500]],ddQ] (* _Harvey P. Dale_, Dec 16 2014 *)

%Y Cf. A068492, A240623.

%K nonn,base

%O 1,1

%A _Michel Lagneau_, Apr 09 2014