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A093324 a(n) is the smallest natural number m such that n^m + m is prime.

%I A093324 #26 Aug 15 2023 02:10:00
%S A093324 2,1,1,2,1,7954,1,34,101,2,1,1181716,1,54,17,2,1,1080,1,57910,9,2,1,
%T A093324 202,2075,5538,3
%N A093324 a(n) is the smallest natural number m such that n^m + m is prime.
%C A093324 a(n)=1 iff n+1 is prime. For n>1 n+a(n) is odd (for n>1 a(n) is odd iff n is even). a(11) is greater than 7000. a(m) for m = 12, 13, 14, 15, 16, 17 and 18 are 1, 54, 17, 2, 1, 1080, 1. 5^7954 + 7954 is a 5560-digit probable prime.
%C A093324 a(11) is greater than 8651. - _Robert G. Wilson v_, May 24 2004
%C A093324 a(11) is greater than 20165, so every prime of the form 11^m + m has more than 21000 digits. - _Farideh Firoozbakht_, Jun 10 2004
%C A093324 a(11) is greater than 80000. - _T. D. Noe_, Mar 07 2007
%C A093324 a(11) is greater than 190000. - _T. D. Noe_, Jul 31 2008
%H A093324 Henri Lifchitz & Renaud Lifchitz, <a href="http://www.primenumbers.net/prptop/searchform.php?form=11%5E1181716%2B1181716">11^1181716+1181716</a>, a(11).
%H A093324 Henri Lifchitz & Renaud Lifchitz, <a href="http://www.primenumbers.net/prptop/searchform.php?form=19%5E57910%2B57910">19^57910+57910</a>, a(19).
%e A093324 a(3)=2 because 3^2 + 2 is prime and 3^1 + 1 is composite.
%t A093324 a[n_]:= (For[m=1, !PrimeQ[n^m+m], m++]; m); Do[Print[a[n]], {n,0,10}]
%o A093324 (Python)
%o A093324 from sympy import isprime
%o A093324 def a(n):
%o A093324   m = 0
%o A093324   while not isprime(n**m + m): m += 1
%o A093324   return m
%o A093324 for n in range(11):
%o A093324   print(a(n), end=", ") # _Michael S. Branicky_, Feb 01 2021
%o A093324 (Magma)
%o A093324 function A093324(n)
%o A093324   t:=0;
%o A093324     while not IsPrime(n^t + t) do
%o A093324       t+:=1;
%o A093324     end while;
%o A093324   return t;
%o A093324 end function;
%o A093324 [A093324(n): n in [0..10]]; // _G. C. Greubel_, Aug 10 2023
%K A093324 nonn,hard,more
%O A093324 0,1
%A A093324 _Farideh Firoozbakht_, May 11 2004
%E A093324 a(11)-a(26) from _Kellen Shenton_, Aug 14 2023