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A301427 Least nonnegative integer k such that n! - n - k is prime.

%I A301427 #41 Oct 29 2020 02:41:30
%S A301427 0,1,2,5,10,23,4,1,2,1,10,3,32,37,42,23,82,11,10,51,66,49,124,11,16,
%T A301427 73,2,49,30,131,14,159,78,91,60,41,34,43,90,37,66,65,8,43,32,55,10,47,
%U A301427 128,15,6,73,6,405,220,51,78,79,10,9,38,295,62,251,124,183,34,27,680,91,300
%N A301427 Least nonnegative integer k such that n! - n - k is prime.
%C A301427 The (n-1) consecutive numbers n!-n, ... , n!-2 (for n > 3) are not prime.
%H A301427 Seiichi Manyama, <a href="/A301427/b301427.txt">Table of n, a(n) for n = 3..500</a>
%F A301427 a(n) = A037155(n) - n.
%e A301427 a(3)=0 because 3! - 3 - 0 =   3 is prime.
%e A301427 a(4)=1 because 4! - 4 - 1 =  19 is prime and 20 is not.
%e A301427 a(5)=2 because 5! - 5 - 2 = 113 is prime and 114 and 115 are not prime.
%p A301427 f:= proc(n) local r; r:= n!-n;
%p A301427   r - prevprime(r)
%p A301427 end proc:
%p A301427 f(3):= 0:
%p A301427 seq(f(i),i=3..100); # _Robert Israel_, Mar 23 2018
%t A301427 a[n_] := n! - NextPrime[n! - 1, -1] - n;
%t A301427 a /@ Range[3, 100] (* _Jean-François Alcover_, Oct 26 2020 *)
%o A301427 (PARI) a(n) = apply(x->(x-precprime(x)), n!-n);
%o A301427 vector(99, n, a(n+2)) \\ _Altug Alkan_, Mar 21 2018
%Y A301427 Cf. A037155, A073443, A090786.
%K A301427 nonn
%O A301427 3,3
%A A301427 _Seiichi Manyama_, Mar 21 2018