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

%I A058814 #40 Feb 16 2025 08:32:43
%S A058814 1,22,23,24,266,267,268,2712,2713,27175,27176,271819,271820,271821,
%T A058814 2718272,2718273,27182807,27182808,271828170,271828171,271828172
%N A058814 Numbers k such that k divides the number of digits of k!.
%C A058814 For k = 1, 22, 23 and 24 only, the number of digits in k! is equal to k. - _Bernard Schott_, Feb 02 2013
%C A058814 I employed R. Wm. Gosper's approximation (A090583). - _Robert G. Wilson v_, Feb 04 2013
%C A058814 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
%D A058814 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 A058814 D. Wells, Curious and Interesting Numbers, Penguin Books, 1997, page 78.
%H A058814 Robert G. Wilson v, <a href="/A058814/b058814.txt">Table of n, a(n) for n = 1..102</a>
%H A058814 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StirlingsApproximation.html">Stirling's Approximation</a>
%e A058814 23! = 25852016738884976640000 has 23 digits.
%t A058814 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 *)
%o A058814 (PARI)
%o A058814 A034886(n)= /* Number of digits in n! */;
%o A058814 { if(n==0, 1, 1 + floor((-n + (2*n+1)*log(n)/2 + 1/2*log(2*Pi))/log(10)) + (n==1)); }
%o A058814 goA058814(maxsearch)= /* write b-File for A058814 */
%o A058814 { my(k=0); for(n=1, maxsearch, if(A034886(n)%n==0, k++; print(k" "n);write("b058814.txt",k" "n);));}
%o A058814 /* _Enrique Pérez Herrero_, Jun 05 2011 */
%Y A058814 Cf. A000142, A034886, A061010. - _Enrique Pérez Herrero_, Jun 05 2011
%K A058814 nonn,base
%O A058814 1,2
%A A058814 _Robert G. Wilson v_, Jan 03 2001