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Smallest number m such that the concatenation of n+1 numbers m^0, m^1,..., m^(n-1), m^n is a prime.
1

%I #20 Aug 08 2020 05:50:18

%S 1,3,33,23,237,93,37,291,421,7,471,213,1941,29,43,17,327,1,523,21,

%T 3403,1,13281,3851,3583,3081,1681,157,34989,5411,2431,1229,4767,1021,

%U 8397,603,429,561,6571,47,8601,347,15027,687,1611,273,29979,201,5719

%N Smallest number m such that the concatenation of n+1 numbers m^0, m^1,..., m^(n-1), m^n is a prime.

%C Conjecture: This sequence is infinite. a(n) = 1 if and only if n + 1 is in the sequence A004023, so a(1) = a(18) = a(22) = a(316) = a(1030) = a(49080) = a(86452) = 1 and there are no other n less than 86453 such that a(n) = 1. Every term of this sequence is odd and for each n, 5 doesn't divide a(n). a(50) is greater than 11111.

%e a(2) = 3 because 3^0 = 1, 3^1 = 3, and 3^2 = 9; 139 is prime and 3 is the smallest number that gives this result.

%e a(3) = 33 because 33^0 = 1; 33^1 = 33; 33^2 = 1089; 33^3 = 35937; 133108935937 is a prime and 33 is the smallest number that gives this result.

%t snm[n_]:=Module[{m=1},While[!PrimeQ[FromDigits[Flatten[ IntegerDigits/@ (m^Range[0,n])]]], m++];m]; Array[snm,50] (* _Harvey P. Dale_, Aug 10 2016 *)

%Y Cf. A004023, A213072.

%K base,nonn

%O 1,2

%A _Farideh Firoozbakht_, Jun 23 2004

%E First example corrected by _Harvey P. Dale_, Aug 10 2016