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A301431 Least nonnegative integer k such that (n!)^2 + n + k + 1 is prime.

%I A301431 #38 Mar 22 2018 09:44:54
%S A301431 0,0,0,1,6,1,4,3,4,13,6,1,46,9,16,7,24,41,48,9,10,81,366,35,82,21,100,
%T A301431 39,152,71,66,377,4,27,8,25,10,225,70,13,158,125,294,3,86,81,26,133,
%U A301431 208,141,50,31,26,127,112,173,802,363,374,47,910,437,74,213,1044,13,1962,41,160,169,296,29
%N A301431 Least nonnegative integer k such that (n!)^2 + n + k + 1 is prime.
%C A301431 The (n-1) consecutive numbers (n)^2! + 2, ..., (n!)^2 + n (for n >= 2) are not prime powers (cf. A246655).
%H A301431 Seiichi Manyama, <a href="/A301431/b301431.txt">Table of n, a(n) for n = 0..500</a>
%e A301431 a(0)=0 because (0!)^2 + 0 + 0 + 1 =   2 is prime.
%e A301431 a(1)=0 because (1!)^2 + 1 + 0 + 1 =   3 is prime.
%e A301431 a(2)=0 because (2!)^2 + 2 + 0 + 1 =   7 is prime.
%e A301431 a(3)=1 because (3!)^2 + 3 + 1 + 1 =  41 is prime and 40 is not prime.
%e A301431 a(4)=6 because (4!)^2 + 4 + 6 + 1 = 587 is prime and 581, 582, ... , 586 are not prime.
%o A301431 (PARI) a(n) = apply(x->(nextprime(x)-x), (n!)^2+n+1); \\ _Altug Alkan_, Mar 21 2018
%Y A301431 Cf. A046029, A090786, A246655.
%K A301431 nonn
%O A301431 0,5
%A A301431 _Seiichi Manyama_, Mar 21 2018