A004152 Sum of digits of n!.

1, 1, 2, 6, 6, 3, 9, 9, 9, 27, 27, 36, 27, 27, 45, 45, 63, 63, 54, 45, 54, 63, 72, 99, 81, 72, 81, 108, 90, 126, 117, 135, 108, 144, 144, 144, 171, 153, 108, 189, 189, 144, 189, 180, 216, 207, 216, 225, 234, 225, 216, 198, 279, 279, 261, 279, 333, 270, 288

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• base

Comments

If n > 5, then 9 divides a(n). - Enrique Pérez Herrero, Mar 01 2009

Examples

			a(5) = 3 because 5! = 120 and 1 + 2 + 0 = 3.
a(6) = 9 because 6! = 720 and 7 + 2 + 0 = 9.
		

Links

• Maciej Ireneusz Wilczynski, Table of n, a(n) for n = 0..10000
• Florian Luca, The number of non-zero digits of n!, Canad. Math. Bull. 45 (2002), pp. 115-118.
• Carlo Sanna, On the sum of digits of the factorial, Journal of Number Theory 147 (February 2015), pp. 836-841. arXiv:1409.4912 [math.NT].
• Carlo Sanna, On the sum of digits of the factorial, Journal of Number Theory 147 (February 2015), pp. 836-841.

Crossrefs

Cf. A000142 (factorial), A007953 (sum of digits), A079584 (same in base 2), A086358 (digital root of n!).

Cf. A066419 (k such that a(k) does not divide k!).

Programs

Magma

[&+Intseq(Factorial(n)): n in [0..70]]; // Vincenzo Librandi, Jan 30 2015

Maple

seq(convert(convert(n!,base,10),`+`),n=0..100); # Robert Israel, Nov 13 2014

Mathematica

Table[ Plus @@ IntegerDigits[n!], {n, 0, 100}] (* Enrique Pérez Herrero, Mar 01 2009 *)

PARI

a(n)=my(v=eval(Vec(Str(n!))));sum(i=1,#v,v[i]) \\ Charles R Greathouse IV, Dec 27 2011

PARI

a(n) = sumdigits(n!); \\ Michel Marcus, Sep 18 2014

Formula

Luca shows that a(n) >> log n. In particular, a(n) > log_10 n - log_10 log_10 n. - Charles R Greathouse IV, Dec 27 2011

a(n) < floor(log_10(n)*9/2). - Carmine Suriano, Feb 20 2013

a(n) = A007953(A000142(n)). - Michel Marcus, Sep 18 2014

a(n) < 9*(A034886(n) - A027868(n)). - Enrique Pérez Herrero, Nov 16 2014

Sanna improved Luca's result to a(n) >> log n log log log n. - Charles R Greathouse IV, Jan 30 2015

a(n) = 9*A202708(n), n>=6. - R. J. Mathar, Jul 30 2021