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 A006488 Numbers n such that n! has a square number of digits. (Formerly M0830) 3

%I M0830

%S 0,1,2,3,7,12,18,32,59,81,105,132,228,265,284,304,367,389,435,483,508,

%T 697,726,944,1011,1045,1080,1115,1187,1454,1494,1617,1659,1788,1921,

%U 2012,2105,2200,2248,2395,2445,2861,2915,3192,3480,3539,3902,3964,4476

%N Numbers n such that n! has a square number of digits.

%C Numbers whose square is represented by the number of digits of n!: 1, 2, 3, 4, 6, 9, 11, 13, 15, 21, 23, 24, 25, 28, 29, ..., . - _Robert G. Wilson v_, May 14 2014

%D M. Gardner, Mathematical Magic Show. Random House, NY, 1978, p. 55.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Robert G. Wilson v, <a href="/A006488/b006488.txt">Table of n, a(n) for n = 0..1310</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StirlingsApproximation.html">Stirling's Approximation</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StirlingsSeries.html">Stirling's Series</a>

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>

%t 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)]]; Select[ Range[ 4500], IntegerQ[ Sqrt[ (LogBase10Stirling[ # ] + 1)]] & ] (* The Mathematica coding comes from J. Stirling's expansion for the Gamma function; see the links. For more terms inside the last Log_10 function, use A001163 & A001164. _Robert G. Wilson v_, Apr 27 2014 *)

%o (PARI) isok(n) = issquare(#Str(n!)); \\ _Michel Marcus_, Sep 05 2015

%Y Cf. A000142, A056851.

%K nonn,base

%O 1,3

%A _N. J. A. Sloane_ and _Robert G. Wilson v_, Jul 11 1991

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