A004152 -id:A004152 - cp's OEIS frontend

A116988 Sum of digits of (10^n)!.

1, 27, 648, 10539, 149346, 1938780, 23903442, 284222502, 3292100235, 37420852599

Offset: 0

Zak Seidov, Apr 02 2006

Cf. A004152 Sum of digits of factorial numbers, A113364 1,27,648,16245,... with first three terms coinciding with this SEQ.

			a(1)=27 because 10!=3628800 and 3+6+2+8+8+0+0=27.

Cf. A004152, A113364.

Do[Print[Total[IntegerDigits[(10^n)! ]]], {n, 0, 7}]

from math import factorial
def A116988(n):
    return sum(int(d) for d in str(factorial(10**n))) # Chai Wah Wu, May 21 2018

One more term from Ryan Propper, Jun 27 2007

a(9) from Chai Wah Wu, May 21 2018

A127554 Sum of the digits of left factorial !n.

Original entry on oeis.org

0, 1, 2, 4, 1, 7, 10, 19, 19, 19, 19, 28, 37, 37, 55, 55, 37, 46, 46, 73, 73, 64, 82, 100, 100, 118, 109, 100, 127, 127, 145, 118, 163, 145, 154, 172, 172, 154, 181, 181, 199, 172, 226, 208, 253, 226, 262, 262, 253, 271, 235, 271, 262, 280, 325, 325, 307, 343, 334

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Apr 17 2007

			!5 = 0!+1!+2!+3!+4! = 1+1+2+6+24 = 34 --> 3+4=7.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A003422, A004152, A007953.

P:=proc(n) local a,i,k,w; a:=0; for i from 1 by 1 to n do w:=0;k:=(i-1)!+a; a:=k; while k>0 do w:=w+k-(trunc(k/10)*10); k:=trunc(k/10); od; print(w); od; end: P(100);

Array[Total[IntegerDigits[Total[Range[0,#-1]!]]]&,59,0] (* James C. McMahon, Jan 01 2025 *)

a(n) = A007953(A003422(n)). - R. J. Mathar, Apr 22 2007

A202708 Sum of digits of n! divided by 9.

Original entry on oeis.org

1, 1, 1, 3, 3, 4, 3, 3, 5, 5, 7, 7, 6, 5, 6, 7, 8, 11, 9, 8, 9, 12, 10, 14, 13, 15, 12, 16, 16, 16, 19, 17, 12, 21, 21, 16, 21, 20, 24, 23, 24, 25, 26, 25, 24, 22, 31, 31, 29, 31, 37, 30, 32, 36, 32, 35, 34, 37, 36, 39, 39, 41, 38, 39, 51, 47, 48, 35, 42, 48

Offset: 6

Michel Lagneau, Dec 23 2011

(sum of digits of n!) / 9 is an integer for n > 5.

Seiichi Manyama, Table of n, a(n) for n = 6..10000

Cf. A004152, A086358.

Table[Sum[DigitCount[n!][[i]]*i/9, {i, 1, 9}], {n, 6, 100}]
Total[IntegerDigits[#]]/9&/@(Range[6,80]!) (* Harvey P. Dale, Nov 22 2023 *)

a(n) = sumdigits(n!)/9; \\ Michel Marcus, Aug 12 2022

a(n) = A004152(n)/9.

A229024 a(n) is the minimum distance to n! for the sum-of-digits of any factorial.

Original entry on oeis.org

0, 0, 0, 3, 3, 9, 27, 18, 0, 0, 9, 9, 0

Offset: 1

Hans Havermann, Sep 11 2013

One could talk of signed integers here: 0, 0, 0, +3, -3, +9, +27, -18, 0, 0, +9, +9, depending on whether the minimum sum-of-digits finds itself above (plus) or below (minus) n!. The problem with so doing is that there might exist some n for which a nonzero minimum distance is both plus and minus.
Zeros indicate where there are solutions in A228311.
List of solutions:
1!    0  (0, 1)
2!    0  (2)
3!    0  (3, 4)
4!   +3  (9, 10, 12, 13)
5!   -3  (30)
6!   +9  (116)
7!  +27  (541, 554)
8!  -18  (3154, 3186, 3219)
9!    0  (21966)
10!   0  (176755)
11!  +9  (1607130)
12!  +9  (16305323)
13!   0  (182624820)

			The minimum distance to 4! is 3, given by the sum of digits for 9!, 10!, 12!, or 13!.
The minimum distance to 5! is also 3, given by the sum of digits of 30!.

Hans Havermann, Determination of a(11)
Hans Havermann, Determination of a(12)
Hans Havermann, Determination of a(13)

Cf. A004152, A228311.

a(13) from Hans Havermann, Nov 04 2013

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

Original entry on oeis.org

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

Offset: 0

Jon E. Schoenfield, Mar 28 2018

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

			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.

Index entries for sequences related to factorial numbers

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!)!).

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

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

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

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

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

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

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

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

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

A228311 Numbers k such that the sum of digits of k! is itself a factorial.

Original entry on oeis.org

0, 1, 2, 3, 4, 21966, 176755, 182624820

Offset: 1

Charles R Greathouse IV, Aug 27 2013

The sum of digits of k! is approximately (9/2)*(d-z), where d=A034886(k) is the number of digits of k!, which is about (log(k/E)*k + log(2*k*Pi)/2)/log(10), and z=A027868(k) is the number of trailing zeros of k!, which is Sum_{j>=1} floor(k/5^j). - Giovanni Resta, Aug 28 2013
a(9) > 2.235*10^9. - Hans Havermann, May 16 2014
k! has ~ k log_10(k) digits, so its digit sum is typically close to C*k*log_10(k) for some constant C. A random number around j has probability something like log(j)/(j log log(j)) of being a factorial, so the probability that the digit sum of k! is a factorial should be something like const/(k log log k). The sum of this diverges, so we should expect infinitely many terms in the sequence. - Robert Israel, Aug 08 2014

			The sum of the digits of 21966! is 362880 = 9!.
The sum of the digits of 176755! is 3628800 = 10!.
The sum of the digits of 182624820! is 6227020800 = 13!.

Shyam Sunder Gupta, Fascinating Factorials, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 16, 411-442.
"Mouhaha" Digit sums and factorials

Cf. A229024, A004152.

lst = {0}; k = p = 1; fctl = Range@ 15!; While[k < 180000, p = p*k; While[ Mod[p, 10] == 0, p /= 10]; If[ MemberQ[ fctl, Plus @@ IntegerDigits@ p], Print[k]; AppendTo[lst, k]]; k++]; lst (* Robert G. Wilson v, Feb 18 2014 *)
With[{fcts=Range[20]!},Select[Range[0,22000],MemberQ[fcts,Total[IntegerDigits[#!]]]&]] (* Harvey P. Dale, Jan 06 2024 *)

lpf(n)=my(f=factor(n)[,1]); f[1]
factorial_lval(n, p)={
    my(s);
    while(n\=p, s+=n);
    s
};
isfactorial(n)={
    if(n<3, return(n>0));
    my(v2=valuation(n,2),mn=v2+1,mx=mn+log(v2+1.5)\log(2),t,c);
    while (mx - mn > 1,
        t = mn + (mx - mn)\2;
        c = factorial_lval(t, 2);
        if (c < v2,
            mn = t+1
        ,
            if (c > v2,
                mx = t-1
            ,
                mx = bitor(t,1);
                mn = max(mn, mx-1)
            )
        )
    );
    if (mn < mx,
        my(p=lpf(mx));
        t = valuation(n, p);
        c = factorial_lval(mx, p);
        if (t > c,return(0));
        if (t == c,
            mn = mx
        )
    );
    n==mn!
};
is(n)=isfactorial(sumdigits(n!))

a(8) from Hans Havermann, Mar 24 2014

A304335 Sum of digits of (2*n-1)!!.

Original entry on oeis.org

1, 1, 3, 6, 6, 18, 18, 18, 18, 36, 45, 45, 36, 63, 72, 72, 90, 90, 108, 108, 126, 108, 144, 144, 135, 144, 153, 180, 180, 171, 180, 198, 198, 189, 216, 243, 243, 234, 252, 225, 243, 261, 252, 297, 306, 333, 360, 324, 342, 342, 315, 360, 369, 378, 396, 387, 387, 414

Offset: 0

Seiichi Manyama, May 11 2018

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A001147, A004152.

a:= n-> add(i, i=convert(doublefactorial(2*n-1), base, 10)):
seq(a(n), n=0..60);  # Alois P. Heinz, May 11 2018

PARI
```
{a(n) = sumdigits((2*n)!/(n!*2^n))}
```

a(n) = A007953(A001147(n)).

A349403 Sum of the digits of Sum_{k=1..n} k!.

Original entry on oeis.org

0, 1, 3, 9, 6, 9, 18, 18, 18, 18, 27, 36, 36, 54, 54, 36, 45, 45, 72, 72, 63, 81, 99, 99, 117, 108, 99, 126, 126, 144, 117, 162, 144, 153, 171, 171, 153, 180, 180, 198, 171, 225, 207, 252, 225, 261, 261, 252, 270, 234, 270, 261, 279, 324, 324, 306, 342, 333, 333, 297, 351, 360, 351, 333, 387, 405, 369

Offset: 0

Seiichi Manyama, Nov 15 2021

If n > 4, then 9 divides a(n).

			   n |  A007489 | A003422 | a(n) | A127554 |
-----+----------+---------+------+---------+
   0 |        0 |       0 |    0 |       0 |
   1 |        1 |       1 |    1 |       1 |
   2 |        3 |       2 |    3 |       2 |
   3 |        9 |       4 |    9 |       4 |
   4 |       33 |      10 |    6 |       1 |
   5 |      153 |      34 |    9 |       7 |
   6 |      873 |     154 |   18 |      10 |
   7 |     5913 |     874 |   18 |      19 |
   8 |    46233 |    5914 |   18 |      19 |
   9 |   409113 |   46234 |   18 |      19 |
  10 |  4037913 |  409114 |   27 |      19 |
  11 | 43954713 | 4037914 |   36 |      28 |

Cf. A004152, A007489, A007953, A127554.

Table[Total[IntegerDigits[Sum[k!,{k,n}]]],{n,0,66}] (* Stefano Spezia, Nov 16 2021 *)

PARI
```
a(n) = sumdigits(sum(k=1, n, k!));
```

a(n) = A007953(A007489(n)).

a(n) = A127554(n+1) - 1 for n > 3.

A202706 Numbers k such that (sum of digits of k!) / 9 is prime.

Original entry on oeis.org

9, 10, 12, 13, 14, 15, 16, 17, 19, 21, 23, 30, 36, 37, 45, 52, 53, 54, 55, 56, 63, 67, 71, 78, 82, 84, 88, 89, 94, 98, 101, 106, 109, 110, 124, 126, 127, 131, 132, 137, 139, 141, 144, 146, 170, 175, 195, 199, 224, 255, 263, 267, 270, 271, 276, 277, 278, 281

Offset: 1

Michel Lagneau, Dec 23 2011

Numbers k such that A004152(k)/9 is prime.

If k > 5, then (sum of digits of k!) / 9 is an integer (see A004152).

			10 is in the sequence because 10!= 3628800 and (3+6+2+8+8)/9 = 27/9 = 3 is prime.

Cf. A000142, A004152.

lst={}; Do[If[PrimeQ[Sum[DigitCount[n!][[i]]*i/9,{i,1,9}]], AppendTo[lst, n]], {n, 300}]; lst

A242538 Squares that are sum of digits of factorials.

Original entry on oeis.org

1, 9, 36, 81, 144, 225, 324, 441, 729, 1089, 1296, 2025, 2304, 2601, 2916, 3249, 3600, 3969, 4356, 4761, 6561, 7056, 7569, 8100, 8649, 9216, 9801, 10404, 11025, 11664, 12996, 13689, 15129, 16641, 17424, 20736, 22500, 23409, 26244, 29241

Offset: 1

Carmine Suriano, May 17 2014

Intersection of A000290 and A004152.

			a(5)=144 for 33!=8683317618811886495518194401280000000 whose sum of digits is 144=12^2. a(5) is also originated from 34! and 35!.

Cf. A066235.

Union[Select[Total[IntegerDigits[#]]&/@(Range[2500]!),IntegerQ[Sqrt[#]]&]] (* Harvey P. Dale, Feb 20 2015 *)

lista(nn) = {v = vector(nn, n, sumdigits(n!)); Set(select(x->issquare(x), v));} \\ Michel Marcus, May 18 2014

cp's OEIS Frontend

A116988 Sum of digits of (10^n)!.

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A127554 Sum of the digits of left factorial !n.

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A202708 Sum of digits of n! divided by 9.

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A229024 a(n) is the minimum distance to n! for the sum-of-digits of any factorial.

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

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Magma

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A228311 Numbers k such that the sum of digits of k! is itself a factorial.

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A304335 Sum of digits of (2*n-1)!!.

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