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A151918 a(n) = k! - prime(n) where k is the smallest number for which prime(n) <= k!.

%I A151918 #10 Mar 17 2019 03:45:42
%S A151918 0,3,1,17,13,11,7,5,1,91,89,83,79,77,73,67,61,59,53,49,47,41,37,31,23,
%T A151918 19,17,13,11,7,593,589,583,581,571,569,563,557,553,547,541,539,529,
%U A151918 527,523,521,509,497,493,491,487,481,479,469,463,457,451,449,443,439,437
%N A151918 a(n) = k! - prime(n) where k is the smallest number for which prime(n) <= k!.
%C A151918 How many times does each prime number appear in this sequence?
%C A151918 Are there infinitely many solutions of the form
%C A151918 (k!-p(n)) = p(r_1)*...*p(r_i); r_i < n for all i?
%F A151918 a(n) = A048765(prime(n)) - prime(n). - _R. J. Mathar_, Aug 25 2010
%e A151918 a(1)  = 2! - p(1)  =   2 -  2 =  0;
%e A151918 a(2)  = 3! - p(2)  =   6 -  3 =  3;
%e A151918 a(3)  = 3! - p(3)  =   6 -  5 =  1;
%e A151918 a(4)  = 4! - p(4)  =  24 -  7 = 17;
%e A151918 a(5)  = 4! - p(5)  =  24 - 11 = 13;
%e A151918 a(6)  = 4! - p(6)  =  24 - 13 = 11;
%e A151918 a(7)  = 4! - p(7)  =  24 - 17 =  7;
%e A151918 a(8)  = 4! - p(8)  =  24 - 19 =  5;
%e A151918 a(9)  = 4! - p(9)  =  24 - 23 =  1;
%e A151918 a(10) = 5! - p(10) = 120 - 29 = 91;
%e A151918 etc.
%p A151918 A048765 := proc(n) for i from 1 do if i! >= n then return i! ; end if; end do: end proc:
%p A151918 A151918 := proc(n) p := ithprime(n) ; A048765(p)-p ; end proc:
%p A151918 seq(A151918(n),n=1..80) ; # _R. J. Mathar_, Aug 25 2010
%t A151918 Module[{fs=Range[10]!,p},Join[{0},Flatten[Table[p=Prime[n];Select[ fs,#>p&,1]-p,{n,2,70}]]]] (* _Harvey P. Dale_, Oct 04 2013 *)
%Y A151918 Cf. A000040, A000142.
%K A151918 easy,nonn
%O A151918 0,2
%A A151918 _Ctibor O. Zizka_, Apr 06 2008
%E A151918 More terms from _R. J. Mathar_, Aug 25 2010