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A004152
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Sum of digits of n!.
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22
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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
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OFFSET
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0,3
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COMMENTS
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If n > 5, then 9 divides a(n). - Enrique Pérez Herrero, Mar 01 2009
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LINKS
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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.
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FORMULA
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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
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EXAMPLE
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a(5) = 3 because 5! = 120 and 1 + 2 + 0 = 3.
a(6) = 9 because 6! = 720 and 7 + 2 + 0 = 9.
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MAPLE
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P:=proc(n) local i, t1, t2; for i from 0 by 1 to n do t1:=i!; t2:=0; while t1 <> 0 do t2:= t2+(t1 mod 10); t1 := floor(t1/10); od; print(t2); od; end: P(100); # Paolo P. Lava, Jun 30 2006
# Simpler alternative:
seq(convert(convert(n!, base, 10), `+`), n=0..100); # Robert Israel, Nov 13 2014
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MATHEMATICA
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Table[ Plus @@ IntegerDigits[n!], {n, 0, 100}] (* Enrique Pérez Herrero, Mar 01 2009 *)
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PROG
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(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
(MAGMA) [&+Intseq(Factorial(n)): n in [0..70]]; // Vincenzo Librandi, Jan 30 2015
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CROSSREFS
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Cf. A000142 (factorial), A007953 (sum of digits), A079584 (same in base 2), A086358 (digital root of n!).
Sequence in context: A105815 A136696 A086358 * A071678 A316164 A319276
Adjacent sequences: A004149 A004150 A004151 * A004153 A004154 A004155
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KEYWORD
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nonn,base
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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