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

1,2

Comments

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

References

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.

Links

Robert G. Wilson v, Table of n, a(n) for n = 1..102

Eric Weisstein's World of Mathematics, Stirling's Approximation

Example

23! = 25852016738884976640000 has 23 digits.

Mathematica

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 *)

Prog

(PARI)
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 */

Crossrefs

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

Sequence in context: A004511 A296763 A236404 * A034895 A106582 A092619

Adjacent sequences: A058811 A058812 A058813 * A058815 A058816 A058817

Keyword

nonn,base

Author

Robert G. Wilson v, Jan 03 2001

Status

approved