A034886 -id:A034886 - cp's OEIS frontend

A058814 Numbers k such that k divides the number of digits of k!.

1, 22, 23, 24, 266, 267, 268, 2712, 2713, 27175, 27176, 271819, 271820, 271821, 2718272, 2718273, 27182807, 27182808, 271828170, 271828171, 271828172

Offset: 1

Robert G. Wilson v, Jan 03 2001

For k = 1, 22, 23 and 24 only, the number of digits in k! is equal to k. - Bernard Schott, Feb 02 2013
I employed R. Wm. Gosper's approximation (A090583). - Robert G. Wilson v, Feb 04 2013
For large m, 10^m*C -> 10^m*e, where e is Euler's or Napier's constant (A001113). Conjecture: There exist at least two contiguous terms for each k > 0, sometimes three contiguous terms, but never four. - Robert G. Wilson v, Feb 04 2013

			23! = 25852016738884976640000 has 23 digits.

Gardner, M. "Factorial Oddities." Ch. 4 in Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind from Scientific American. New York: Vintage, pp. 50-65, 1978
D. Wells, Curious and Interesting Numbers, Penguin Books, 1997, page 78.

Robert G. Wilson v, Table of n, a(n) for n = 1..102
Eric Weisstein's World of Mathematics, Stirling's Approximation

Cf. A000142, A034886, A061010. - Enrique Pérez Herrero, Jun 05 2011

fQ[n_] := Mod[ Floor[(n*Log[n] - n + Log[(2 n + 1/3) Pi]/2)/Log[10] + 1], n] == 0; k = 1; s = {}; While[k < 1000001, If[ fQ@ k, AppendTo[s, k]; Print[k]]; k++]; s (* Robert G. Wilson v, Feb 04 2013 *)

A034886(n)= /* Number of digits in n! */;
{ if(n==0, 1, 1 + floor((-n + (2*n+1)*log(n)/2 + 1/2*log(2*Pi))/log(10)) + (n==1)); }
goA058814(maxsearch)= /* write b-File for A058814 */
{ my(k=0); for(n=1, maxsearch, if(A034886(n)%n==0, k++; print(k" "n);write("b058814.txt",k" "n);));}
/* Enrique Pérez Herrero, Jun 05 2011 */

A063979 Number of decimal digits in (n!)!; A000197.

Original entry on oeis.org

1, 1, 1, 3, 24, 199, 1747, 16474, 168187, 1859934, 22228104, 286078171, 3949867548, 58284826485, 915905054360, 15276520209206, 269617872744249, 5021159048900643, 98417586560408168, 2025488254833817394, 43675043585825292775, 984729344827900257489, 23172929656443132617906

Offset: 0

Robert G. Wilson v, Sep 05 2001

Jon E. Schoenfield, Table of n, a(n) for n = 0..448 (first 100 terms from Robert G. Wilson v)
Eric Weisstein's World of Mathematics, Factorial

Cf. A000197, A063944, A034886.

// Using about 100 more digits of precision than needed.
nMax:=30; SetDefaultRealField(RealField(Ceiling(Log(10,Factorial(nMax))+100))); a:=[]; for n in [0..nMax] do a[n+1]:=1+Floor(LogGamma(Factorial(n)+1)/Log(10)); end for; a; // Jon E. Schoenfield, Aug 07 2015

seq(length((n)!!), n=0..19); # Zerinvary Lajos, Mar 10 2007

LogBase10Stirling[n_] := Floor[ Log[10, 2 Pi n]/2 + n*Log[10, n/E] + Log[10, 1 + 1/(12n) + 1/(288n^2) - 139/(51840n^3) - 571/(2488320n^4) + 163879/(209018880n^5) + 5246819/(75246796800n^6)]]; (* A001163/A001164; good to at least a(1000) *) LogBase10Stirling[0] = LogBase10Stirling[1] = 0; Table[1 + LogBase10Stirling[n!], {n, 0, 101}] (* Robert G. Wilson v, Aug 05 2015 *)

\\ Using 100 digits of precision.
 a(n)=localprec(100); my(t=n!);return(floor((t*log(t)-t+1/2*log(2*Pi*t)+1/(12*t))/log(10)+1))\\ Robert Gerbicz, Jul 08 2008

More terms from Vladeta Jovovic, Sep 06 2001
A correspondent reported that terms a(17) - a(19) shown here were wrong. That's not true, they are correct. The correspondent was using Python, where the default precision was not large enough to calculate these terms correctly. Thanks to Brendan McKay, Max Alekseyev and Robert Gerbicz for confirming the entries. - N. J. A. Sloane, Jul 08 2008
a(20) from Brendan McKay, Jul 08 2008
a(21)-a(22) from Hugo Pfoertner, Nov 25 2023

A333431 Numbers k such that k! has a factorial number of decimal digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 9, 24, 342, 11158, 145435633325318659

Offset: 1

Robert G. Wilson v, Mar 20 2020

From Vaclav Kotesovec, Mar 22 2020: (Start)
a(11) has 1199 digits and a(11)! has 525! digits.
a(12) has 1300 digits and a(12)! has 562! digits.
a(13) has 3733 digits and a(13)! has 1380! digits.
a(14) has 4730 digits and a(14)! has 1693! digits.
a(15), if it exists, must have more than 5732 digits and a(15)! must have more than 2000! digits. (End)

			9 is in the sequence since 9! = 362880 which has 6 decimal digits and 6 = 3!.

Vaclav Kotesovec, a(11)-a(14)

Cf. A000142, A034886.

f = k = 1; lst = {0}; While[k < 12000, f *= k; If[ MemberQ[{1, 2, 6, 24, 120, 720, 5040, 40320, 362880}, IntegerLength@ f], AppendTo[lst, k]]; k++]; lst

a(10) from Giovanni Resta, Mar 21 2020
a(11)-a(12) from Vaclav Kotesovec, Mar 21 2020
a(13)-a(14) from Vaclav Kotesovec, Mar 22 2020

A356758 a(n) is the number of nonzero digits in n!.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 3, 5, 5, 6, 5, 6, 9, 9, 10, 11, 11, 12, 12, 13, 14, 18, 18, 17, 19, 20, 20, 24, 24, 27, 26, 29, 28, 32, 32, 32, 29, 35, 39, 35, 39, 40, 43, 44, 42, 49, 48, 49, 46, 49, 50, 53, 54, 56, 58, 57, 62, 62, 63, 58, 66, 67, 70, 71, 70, 73, 72, 78, 81

Offset: 0

Stefano Spezia, Aug 26 2022

Robert G. Wilson v, Table of n, a(n) for n = 0..10000
Index entries for sequences related to factorial numbers

Cf. A000142, A027869, A034886, A356757.

Table[Length[Select[IntegerDigits[n!],Positive]],{n,0,70}]

a(n) = #select(x->(x>0), digits(n!)); \\ Michel Marcus, Aug 26 2022

from math import factorial
def a(n): return len(str(factorial(n)).replace("0", ""))
print([a(n) for n in range(71)]) # Michael S. Branicky, Aug 26 2022

a(n) = A034886(n!) - A027869(n!).

A008906 Number of digits in n! excluding final zeros.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 3, 4, 5, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 15, 16, 18, 19, 20, 20, 21, 23, 24, 25, 26, 27, 29, 30, 32, 33, 34, 36, 37, 39, 39, 41, 43, 44, 46, 47, 48, 50, 52, 53, 53, 55, 56, 58, 60, 61, 62, 64, 66, 68, 68, 70, 72, 74, 76, 76, 78, 80, 82, 84, 85, 86, 88, 90, 92, 92, 94

Offset: 0

Russ Cox

From Bernard Schott, Nov 19 2021: (Start)
a(n) < n iff 2 <= n <= 38 or n = 40;
a(n) = n iff n = 1, 39, 41;
a(n) > n iff n = 0 or n >= 42. (End)

Cf. A027868, A034886.

Array[IntegerLength[#!//.x_/;x~Mod~10==0:>x/10]&,77,0] (* Giorgos Kalogeropoulos, Nov 19 2021 *)

from math import factorial
def A008906(n): return len(str(factorial(n)).rstrip('0')) # Chai Wah Wu, Oct 24 2021

a(n) = A034886(n) - A027868(n). - Michel Marcus, Jun 24 2013

A104351 Number of digits in decimal representation of A104350(n).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 4, 4, 5, 6, 7, 8, 9, 9, 10, 11, 11, 13, 13, 14, 15, 16, 17, 18, 19, 19, 20, 22, 22, 24, 24, 25, 26, 27, 28, 29, 30, 32, 32, 34, 35, 36, 37, 38, 40, 41, 42, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 54, 55, 57, 58, 59, 59, 60, 61, 63, 64, 66, 67, 69, 69, 71, 72

Offset: 1

Reinhard Zumkeller, Mar 06 2005

Amiram Eldar, Table of n, a(n) for n = 1..10000
Reinhard Zumkeller, Products of largest prime factors of numbers <= n.

Cf. A034886, A055642, A104350.

IntegerLength[FoldList[Times, Array[FactorInteger[#][[-1, 1]] &, 100]]] (* Amiram Eldar, Apr 08 2024 *)

gpf(n) = {my(p = factor(n)[, 1]); p[#p];}
a(n) = logint(prod(k = 2, n, gpf(k)), 10) + 1; \\ Amiram Eldar, Apr 08 2024

a(n) = A055642(A104350(n)).

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

A375348 a(n) is the mode of the digits of n! not counting trailing zeros (using -1 if multimodal).

Original entry on oeis.org

1, 1, 2, 6, -1, -1, -1, -1, -1, 8, 8, 9, 0, 2, -1, -1, 8, -1, 7, -1, -1, -1, 7, 8, 3, 1, 6, 8, -1, -1, 8, 2, 3, 8, 9, -1, 9, -1, 0, 8, 1, -1, -1, 3, 8, 6, -1, 1, 7, 2, 6, -1, 8, 3, -1, 5, 4, 2, -1, 8, 4, 0, 2, 6, -1, 2, 4, 6, 1, 2, 8, 8, 8, 0, 2, 4, -1, 8, 2, 1, 5, 7, 4, -1, 1, 0

Offset: 0

Stefano Spezia and Robert G. Wilson v, Aug 12 2024

Inspired by A356758.
If we were to count trailing zeros, then would have a(n) = 0 for all n >= 34. Therefore we only consider the decimal digits of A004154(n).
Conjecture: excluding -1, as n -> oo, all digits occur equally often.

			a(0) = a(1) = 1 because 0! = 1! = 1 and 1 is the only digit present;
a(4) = -1 since 4! = 24 and there are only two digits appearing with the same frequency, 2 and 4.
a(14) = -1 because 14! = 87178291200 and, not counting the two trailing 0's, there are two 1's, two 2's, two 7's, and two 8's.

Cf. A004154, A027869, A031144, A034886, A061010, A137579, A137580, A356758.

a[n_] := If[Length[c=Commonest[IntegerDigits[n! / 10^IntegerExponent[n!]]]] > 1, -1, c[[1]]]; Array[a, 86, 0]

from collections import Counter
from sympy import factorial
def A375348(n): return -1 if len(k:=Counter(str(factorial(n)).rstrip('0')).most_common(2)) > 1 and k[0][1]==k[1][1] else int(k[0][0]) # Chai Wah Wu, Sep 15 2024

A066026 a(n) = ceiling(log(n!)).

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 9, 11, 13, 16, 18, 20, 23, 26, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 59, 62, 65, 68, 72, 75, 79, 82, 86, 89, 93, 96, 100, 103, 107, 111, 115, 118, 122, 126, 130, 133, 137, 141, 145, 149, 153, 157, 161, 165, 169, 173, 177, 181, 185, 189, 193, 197, 202

Offset: 1

Robert A. Stump (bee_ess107(AT)yahoo.com), Dec 11 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A025201, A050502, A034886.

Ceiling[Log[Range[70]!]] (* Harvey P. Dale, Jul 23 2012 *)

{ for (n=1, 1000, a=ceil(log(n!)); write("b066026.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 07 2009

a(n) = ceil(lngamma(n+1)) \\ Michel Marcus, Jun 29 2015

Terms a(51)-a(63) from Harry J. Smith, Nov 07 2009

A132826 Decimal expansion of the integer Googol!.

Original entry on oeis.org

1, 6, 2, 9, 4, 0, 4, 3, 3, 2, 4, 5, 9, 3, 3, 7, 3, 7, 3, 4, 1, 7, 9, 3, 4, 6, 5, 2, 9, 8, 3, 5, 4, 2, 1, 7, 2, 8, 2, 1, 8, 8, 8, 4, 2, 6, 7, 1, 4, 8, 6, 6, 2, 3, 0, 3, 6, 2, 3, 6, 1, 1, 9, 3, 6, 9, 4, 0, 9, 2, 2, 0, 2, 9, 4, 5, 2, 5, 0, 4, 6, 8, 6, 6, 7, 9, 8, 5, 4, 4, 7, 0, 8, 4, 2, 2, 3, 1, 7, 8, 9, 2, 2, 8, 1

Offset: 1

Martin Raab, Nov 18 2007, Dec 11 2007

The number in question has 9956570551809674817234887108108339491770560299419 \ 63334338855462168341353507911292252707750506615682568 digits and ends in exactly 10^101/8 - 18 zeros. - Robert G. Wilson v, Jan 09 2013

The last nonzero term of this sequence is 6. - Washington Bomfim, Dec 24 2010

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Math.; section 4, exercises 40, and 54.

Cf. A034886, A173670.

f[n_] := 10^FractionalPart[N[(n*Log[n] - n + (1/2)*Log[(2*n + 1/3)*Pi])/Log[10], 203]]; RealDigits[ f[10^100], 10, 101][[1]] (* Robert G. Wilson v, Jan 09 2013 *)

10^100! = 1*2*3*4*...*(10^100-1)*10^100.

cp's OEIS Frontend

A058814 Numbers k such that k divides the number of digits of k!.

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A063979 Number of decimal digits in (n!)!; A000197.

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A333431 Numbers k such that k! has a factorial number of decimal digits.

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A356758 a(n) is the number of nonzero digits in n!.

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A008906 Number of digits in n! excluding final zeros.

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A104351 Number of digits in decimal representation of A104350(n).

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

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