A301861 a(n) is the sum of the decimal digits of (n!)!.

1, 1, 2, 9, 81, 783, 7164, 69048, 711009, 7961040, 95935761, 1242436185, 17235507996

Offset: 0

Author

Jon E. Schoenfield, Mar 28 2018

Keywords

• nonn
• base
• hard
• more

Comments

Presumably, lim_{n->oo} a(n)/A008906(n!) = 9/2.

Examples

			a(0) = digitsum((0!)!) = digitsum(1!) = digitsum(1) = 1.
a(1) = digitsum((1!)!) = digitsum(1!) = digitsum(1) = 1.
a(2) = digitsum((2!)!) = digitsum(2!) = digitsum(2) = 2.
a(3) = digitsum((3!)!) = digitsum(6!) = digitsum(720) = 7+2 = 9.
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.
		

Links

• Index entries for sequences related to factorial numbers

Crossrefs

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!)!).

Programs

Magma

[&+Intseq(Factorial(Factorial(n))): n in [0..10]]; // Vincenzo Librandi, Mar 29 2018

Maple

a:= n-> add(i, i=convert(n!!, base, 10)):
seq(a(n), n=0..8);  # Alois P. Heinz, Oct 27 2021

Mathematica

Table[Plus@@IntegerDigits[(n!)!], {n, 0, 10}] (* Vincenzo Librandi, Mar 29 2018 *)

PARI

a(n) = sumdigits(n!!); \\ Michel Marcus, Mar 28 2018

Python

from math import factorial
def A301861(n):
    return sum(int(d) for d in str(factorial(factorial(n)))) # Chai Wah Wu, Mar 31 2018
# faster program for larger values of n
from gmpy2 import mpz, digits, fac
def A301861(n): return int(sum(mpz(d) for d in digits(fac(fac(n))))) # Chai Wah Wu, Oct 24 2021

Formula

a(n) = A007953(A000197(n)). - Michel Marcus, Mar 28 2018

a(n) = A004152(A000142(n)). - Altug Alkan, Mar 28 2018

Extensions

a(11) from Chai Wah Wu, Mar 31 2018

a(12) from Chai Wah Wu, Apr 01 2018