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%I #86 Aug 26 2024 11:42:56
%S 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,
%T 81,108,90,126,117,135,108,144,144,144,171,153,108,189,189,144,189,
%U 180,216,207,216,225,234,225,216,198,279,279,261,279,333,270,288
%N Sum of digits of n!.
%C If n > 5, then 9 divides a(n). - _Enrique Pérez Herrero_, Mar 01 2009
%H Maciej Ireneusz Wilczynski, <a href="/A004152/b004152.txt">Table of n, a(n) for n = 0..10000</a>
%H Florian Luca, <a href="http://math.ca/10.4153/CMB-2002-013-9">The number of non-zero digits of n!</a>, Canad. Math. Bull. 45 (2002), pp. 115-118.
%H Carlo Sanna, <a href="http://arxiv.org/abs/1409.4912">On the sum of digits of the factorial</a>, Journal of Number Theory 147 (February 2015), pp. 836-841. arXiv:1409.4912 [math.NT].
%H Carlo Sanna, <a href="https://dx.doi.org/10.1016/j.jnt.2014.09.003">On the sum of digits of the factorial</a>, Journal of Number Theory 147 (February 2015), pp. 836-841.
%F 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
%F a(n) < floor(log_10(n)*9/2). - _Carmine Suriano_, Feb 20 2013
%F a(n) = A007953(A000142(n)). - _Michel Marcus_, Sep 18 2014
%F a(n) < 9*(A034886(n) - A027868(n)). - _Enrique Pérez Herrero_, Nov 16 2014
%F Sanna improved Luca's result to a(n) >> log n log log log n. - _Charles R Greathouse IV_, Jan 30 2015
%F a(n) = 9*A202708(n), n>=6. - _R. J. Mathar_, Jul 30 2021
%e a(5) = 3 because 5! = 120 and 1 + 2 + 0 = 3.
%e a(6) = 9 because 6! = 720 and 7 + 2 + 0 = 9.
%p seq(convert(convert(n!,base,10),`+`),n=0..100); # _Robert Israel_, Nov 13 2014
%t Table[ Plus @@ IntegerDigits[n!], {n, 0, 100}] (* _Enrique Pérez Herrero_, Mar 01 2009 *)
%o (PARI) a(n)=my(v=eval(Vec(Str(n!))));sum(i=1,#v,v[i]) \\ _Charles R Greathouse IV_, Dec 27 2011
%o (PARI) a(n) = sumdigits(n!); \\ _Michel Marcus_, Sep 18 2014
%o (Magma) [&+Intseq(Factorial(n)): n in [0..70]]; // _Vincenzo Librandi_, Jan 30 2015
%Y Cf. A000142 (factorial), A007953 (sum of digits), A079584 (same in base 2), A086358 (digital root of n!).
%Y Cf. A066419 (k such that a(k) does not divide k!).
%K nonn,base
%O 0,3
%A _N. J. A. Sloane_