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%I #13 Jul 16 2017 09:38:45
%S 1,9,225,14049,1742049,374990049,128399054049,65676719822049,
%T 47850402559694049,47832576242431694049,63649302669112063694049,
%U 109966989623147836159694049,241567605673714904675071694049
%N a(1) = 1; a(n) = a(n-1) + (n!)^3.
%C By sum of cubes factorization, every a(n) > 1 is a multiple of 9, hence none of these are prime, unlike the case of sum of squares of factorials (i.e. (1!)^2 + (2!)^2+ (3!)^2+ (4!)^2 = 617 is prime; 41117342095090841723228045851817 = (1!)^2 + (2!)^2 + (3!)^2 + (4!)^2 + (5!)^2 + (6!)^2 + (7!)^2 + (8!)^2 + (9!)^2 + (10!)^2 + (11!)^2 + (12!)^2 + (13!)^2 + (14!)^2 + (15!)^2 + (16!)^2 + (17!)^2 + (18!)^2 is prime).
%H Seiichi Manyama, <a href="/A138564/b138564.txt">Table of n, a(n) for n = 1..181</a>
%F a(n) = Sum_{k=1..n} (k!)^3 = Sum_{k=1..n} A000578(A000142(n)).
%e a(18) = (1!)^3 + (2!)^3 + (3!)^3 + (4!)^3 + (5!)^3 + (6!)^3 + (7!)^3 + (8!)^3 + (9!)^3 + (10!)^3 + (11!)^3 + (12!)^3 + (13!)^3 + (14!)^3 + (15!)^3 + (16!)^3 + (17!)^3 + (18!)^3 = 262480797594664584673157017306412926841599694049.
%t nxt[{n_,a_}]:={n+1,a+((n+1)!)^3}; Transpose[NestList[nxt,{1,1},20]][[2]] (* _Harvey P. Dale_, Mar 08 2015 *)
%Y Cf. A000142, A000578, A100288.
%Y Sum_{k=1..n] (k!)^m: A007489 (m=1), A104344 (m=2), this sequence (m=3), A289945 (m=4), A289946 (m=6).
%K easy,nonn
%O 1,2
%A _Jonathan Vos Post_, May 18 2008
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