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%I M0830 #68 Sep 08 2022 08:44:34
%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
%C From _Bernard Schott_, Jan 04 2020: (Start)
%C In M. Gardner's book, see reference, there is a tree printout of 105! with 169 digits, where the bottom row consists of the 25 trailing zeros of 105!. M. Gardner does not explain if this is the only factorial that can be displayed in a similar tree form.
%C Proof: If m! has q^2 digits, hence the number of trailing zeros in m! must be equal to 2*q-1 to satisfy this triangular look; m = 105 satisfies these two conditions with q = 13 because 105! has 13^2 = 169 digits and 2*13-1 = 25 trailing zeros.
%C When m < 105 and m! has q^2 digits (m <= 81), then q <= 11 and the number of trailing zeros is <= 2*q - 3.
%C When m > 105 and m! has q^2 digits (m >= 132), then q >= 15 and the number of trailing zeros is >= 2*q + 2.
%C Hence, only 105! presents such a tree printout.
%C 1
%C 081
%C 39675
%C 8240290
%C 900504101
%C 30580032964
%C 9720646107774
%C 902579144176636
%C 57322653190990515
%C 3326984536526808240
%C 339776398934872029657
%C 99387290781343681609728
%C 0000000000000000000000000
%C (End)
%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 = 1..1311</a>
%H D. S. Kluk and N. J. A. Sloane, <a href="/A002050/a002050_3.pdf">Correspondence, 1979</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StirlingsApproximation.html">Stirling's Approximation</a> and <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 *)
%t Select[Range[0,4500],IntegerQ[Sqrt[IntegerLength[#!]]]&] (* _Harvey P. Dale_, Sep 27 2018 *)
%o (PARI) isok(n) = issquare(#Str(n!)); \\ _Michel Marcus_, Sep 05 2015
%o (Magma) [k:k in [0..5000]| IsSquare(#Intseq(Factorial(k)))]; // _Marius A. Burtea_, Jan 04 2020
%Y Cf. A000142, A027868 (trailing zeros), A034886 (number of digits), A056851.
%K nonn,base
%O 1,3
%A _N. J. A. Sloane_ and _Robert G. Wilson v_
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