A061010 -id:A061010 - cp's OEIS frontend

A214004 Values of n where A061010(n) = A114485(n) + 1.

7, 8, 10, 12, 13, 14, 15, 16, 17, 19, 20, 22, 23, 28, 29, 36, 40, 41, 42, 45, 46, 49, 51, 53, 54, 61, 62, 65, 67, 69, 72, 75, 79, 81, 82, 84, 90, 94, 97, 100, 101, 106, 107, 111, 113, 114, 116, 117, 118, 122, 123, 126, 127, 131, 133, 134, 136, 137, 139, 142, 144, 145, 147, 152, 155, 157, 158, 160, 161, 162, 163, 165, 167, 169, 170, 172, 176, 181, 182, 183, 185, 189, 191, 192, 194

Offset: 1

Carl R. White, Jun 30 2012

Compares numbers of digits between !(10^n) and (10^n)!. At all other positions, A061010 and A114485 are equal.

Cf. A061010, A114485

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

Original entry on oeis.org

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 */

A114485 Number of decimal digits in !(10^n), where !x is the subfactorial.

Original entry on oeis.org

7, 158, 2568, 35660, 456574, 5565709, 65657059, 756570556, 8565705523, 95657055186, 1056570551816, 11565705518103, 125657055180974, 1356570551809682, 14565705518096756, 155657055180967490

Offset: 1

Eric W. Weisstein, Nov 30 2005

Entries where the number of digits of the subfactorial (here) is one less than the number of digits of the factorial (A061010) are indicated by A214004. [Carl R. White, Jun 30 2012]

Carl R. White, Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics, Subfactorial

Cf. A000166, A061010.

a(n) = 10^n*(n - 1/log(10)) + n/2 + O(1). [Arkadiusz Wesolowski, Jan 20 2012]

More terms from Carl R. White, Jun 30 2012

A119906 Largest number whose factorial is less than 10^(10^n).

Original entry on oeis.org

3, 13, 69, 449, 3248, 25205, 205022, 1723507, 14842906, 130202808, 1158787577, 10433891463, 94851898540, 869200494599, 8019346203785, 74419210652835, 694100859679691, 6502464891216879, 61154108320430275, 577134533044522749, 5463531774867094396

Offset: 0

Julien Peter Benney (jpbenney(AT)ftml.net), Aug 01 2006

nonn

			a(1) = 13 because 10^(10^1) = 10^10 = 10000000000 and 13! = 6227020800 is largest factorial less than this.

Daniel Suteu, Table of n, a(n) for n = 0..500

Cf. A000142, A061010, A152909.

A119906Stirling := proc(n) local i; fsolve( (i+0.5)*log(i+1.0)=log(10.0)*10.0^n+i-1.0-0.5*log(2.0*Pi)-1./12.0/i,i) ; end : A119906 := proc(n::integer) local aestim::integer,resul::integer,faci1::integer,faci::integer,pow10::integer,i ; pow10 := 10^(10^n) ; aestim := floor(A119906Stirling(n))-1 ; faci1 := factorial(aestim) ; for i from aestim+1 to aestim+9 do faci := factorial(i) ; if faci1< pow10 and faci >= pow10 then RETURN(i-1) ; fi ; faci1 := faci ; end ; RETURN(0) ; end: for n from 0 to 30 do printf("%d,",A119906(n)) ; od ; # R. J. Mathar, Aug 04 2006

With[{s = Array[Factorial, 30000]}, Table[LengthWhile[s, # < 10^(10^n) &], {n, 0, 5}] ] (* Michael De Vlieger, May 14 2018 *)

a(6) from R. J. Mathar, Aug 04 2006
a(7)-a(19) from Jon E. Schoenfield, Aug 09 2006
a(20) from Daniel Suteu, May 14 2018

A244061 The number of digits of (2^n)!.

Original entry on oeis.org

1, 1, 2, 5, 14, 36, 90, 216, 507, 1167, 2640, 5895, 13020, 28504, 61937, 133734, 287194, 613842, 1306594, 2771010, 5857670, 12346641, 25955890, 54436999, 113924438, 237949763, 496101303, 1032606162, 2146019444, 4453653132, 9230534755

Offset: 0

Robert G. Wilson v, Jun 18 2014

Cf. A000722, A061010, A244060.

LogBase10Stirling[n_] := Floor[ Log[10, 2 Pi n]/2 + n*Log[10, n/E] + Log[10, 1 + 1/(12 n) + 1/(288 n^2) - 139/(51840 n^3) - 571/(2488320 n^4) + 163879/(209018880 n^5)]]; Table[ LogBase10Stirling[2^n] + 1, {n, 0, 30}]
IntegerLength[(2^Range[0,30])!] (* Harvey P. Dale, Nov 05 2021 *)

A261175 Number of digits of Hyperfactorial(n).

Original entry on oeis.org

1, 1, 1, 3, 5, 8, 13, 19, 26, 35, 45, 56, 69, 84, 100, 117, 137, 158, 180, 204, 231, 258, 288, 319, 352, 387, 424, 463, 503, 546, 590, 636, 684, 734, 786, 840, 897, 955, 1015, 1077, 1141, 1207, 1275, 1345, 1418, 1492, 1568, 1647, 1728, 1811, 1896, 1983, 2072, 2163, 2257, 2352

Offset: 0

Robert G. Wilson v, Aug 20 2015

nonn

Chai Wah Wu, Table of n, a(n) for n = 0..1000

Cf. A002109, A061010, A260262.

Table[1 + Floor@ Log10@ Hyperfactorial@ n, {n, 0, 55}]

a(n)=my(r=1);for(i=1,n+1,r *= i^i);floor(log(r)/log(10))+1 \\ Anders Hellström, Aug 20 2015

vector(60, n, n--; #Str(prod(k=2, n, k^k))) \\ Michel Marcus, Aug 20 2015

A261175_list, n = [], 1
for i in range(100):
    n *= i**i
    A261175_list.append(len(str(n)))  # Chai Wah Wu, Aug 21 2015

a(n) = 1 + floor( log_10( A002109(n))) = A055642(A002109(n)).

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

A375575 a(n) is the least frequent digit of n! not counting trailing zeros, or -1 if there is more than one least frequent digit.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia and Robert G. Wilson v, Aug 19 2024

Analogous to A375348.
If we were to count trailing zeros, then a(n) would never equal zero, for all n's >= 0. Therefore we only consider the decimal digits of A004154.
Conjecture: excluding -1, as n -> oo, the digits distribution is uniform as in A375348.

			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 two least frequent digits, 2 and 4.
a(14) = 9 because 14! = 87178291200 and, not counting the two trailing 0's, there are two 1's, two 2's, two 7's, two 8's but only one 9.

Cf. A004154, A027869, A031144, A034886, A061010, A137579, A137580, A375348.

f:= proc(n) local L,j;
  L:= convert(n!,base,10);
  for j from 1 while L[j] = 0 do od:
  L:= Statistics:-Tally(L[j...-1]);
  L:= sort(L,(a,b) -> rhs(a) < rhs(b));
  if nops(L) >= 2 and rhs(L[2]) = rhs(L[1]) then -1 else lhs(L[1]) fi
end proc:
map(f, [$0..100]); # Robert Israel, Sep 02 2024

Rarest[lst_] := MinimalBy[ Tally[lst], Last][[All, 1]]; a[n_] := If[ Length[c = Rarest[ IntegerDigits[n!/10^IntegerExponent[n!, 10]] ]] >1, -1, c[[1]]]; Array[a, 80, 0]

from collections import Counter
from sympy import factorial
def A375575(n): return -1 if len(k:=Counter(str(factorial(n)).rstrip('0')).most_common()) > 1 and k[-1][1]==k[-2][1] else int(k[-1][0]) # Chai Wah Wu, Sep 15 2024

cp's OEIS Frontend

A214004 Values of n where A061010(n) = A114485(n) + 1.

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

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A114485 Number of decimal digits in !(10^n), where !x is the subfactorial.

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A119906 Largest number whose factorial is less than 10^(10^n).

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Maple

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A244061 The number of digits of (2^n)!.

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Mathematica

A261175 Number of digits of Hyperfactorial(n).

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

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A375575 a(n) is the least frequent digit of n! not counting trailing zeros, or -1 if there is more than one least frequent digit.

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Maple

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Python