%I #43 Mar 20 2022 14:29:36
%S 1,1,2,9,81,783,7164,69048,711009,7961040,95935761,1242436185,
%T 17235507996
%N a(n) is the sum of the decimal digits of (n!)!.
%C Presumably, lim_{n->oo} a(n)/A008906(n!) = 9/2.
%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F a(n) = A007953(A000197(n)). - _Michel Marcus_, Mar 28 2018
%F a(n) = A004152(A000142(n)). - _Altug Alkan_, Mar 28 2018
%e a(0) = digitsum((0!)!) = digitsum(1!) = digitsum(1) = 1.
%e a(1) = digitsum((1!)!) = digitsum(1!) = digitsum(1) = 1.
%e a(2) = digitsum((2!)!) = digitsum(2!) = digitsum(2) = 2.
%e a(3) = digitsum((3!)!) = digitsum(6!) = digitsum(720) = 7+2 = 9.
%e a(4) = digitsum((4!)!) = digitsum(24!) = digitsum(620448401733239439360000) = 6+2+0+4+4+8+4+0+1+7+3+3+2+3+9+4+3+9+3+6+0+0+0+0 = 81.
%p a:= n-> add(i, i=convert(n!!, base, 10)):
%p seq(a(n), n=0..8); # _Alois P. Heinz_, Oct 27 2021
%t Table[Plus@@IntegerDigits[(n!)!], {n, 0, 10}] (* _Vincenzo Librandi_, Mar 29 2018 *)
%o (PARI) a(n) = sumdigits(n!!); \\ _Michel Marcus_, Mar 28 2018
%o (Magma) [&+Intseq(Factorial(Factorial(n))): n in [0..10]]; // _Vincenzo Librandi_, Mar 29 2018
%o (Python)
%o from math import factorial
%o def A301861(n):
%o return sum(int(d) for d in str(factorial(factorial(n)))) # _Chai Wah Wu_, Mar 31 2018
%o # faster program for larger values of n
%o from gmpy2 import mpz, digits, fac
%o def A301861(n): return int(sum(mpz(d) for d in digits(fac(fac(n))))) # _Chai Wah Wu_, Oct 24 2021
%Y Cf. A000142 (factorial numbers), A000197 ((n!)!), A004152 (sum of digits of n!), A007953 (sum of digits of n), A008906 (number of digits in n! excluding trailing zeros), A027868 (number of trailing zeros in n!), A034886 (number of digits in n!), A063979 (number of digits in (n!)!).
%K nonn,base,hard,more
%O 0,3
%A _Jon E. Schoenfield_, Mar 28 2018
%E a(11) from _Chai Wah Wu_, Mar 31 2018
%E a(12) from _Chai Wah Wu_, Apr 01 2018