

A006488


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


3



0, 1, 2, 3, 7, 12, 18, 32, 59, 81, 105, 132, 228, 265, 284, 304, 367, 389, 435, 483, 508, 697, 726, 944, 1011, 1045, 1080, 1115, 1187, 1454, 1494, 1617, 1659, 1788, 1921, 2012, 2105, 2200, 2248, 2395, 2445, 2861, 2915, 3192, 3480, 3539, 3902, 3964, 4476
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OFFSET

1,3


COMMENTS

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


REFERENCES

M. Gardner, Mathematical Magic Show. Random House, NY, 1978, p. 55.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

Robert G. Wilson v, Table of n, a(n) for n = 1..1311
D. S. Kluk and N. J. A. Sloane, Correspondence, 1979
Eric Weisstein's World of Mathematics, Stirling's Approximation and Stirling's Series
Index entries for sequences related to factorial numbers


MATHEMATICA

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 *)
Select[Range[0, 4500], IntegerQ[Sqrt[IntegerLength[#!]]]&] (* Harvey P. Dale, Sep 27 2018 *)


PROG

(PARI) isok(n) = issquare(#Str(n!)); \\ Michel Marcus, Sep 05 2015


CROSSREFS

Cf. A000142, A056851.
Sequence in context: A302506 A228828 A061577 * A121430 A217379 A023606
Adjacent sequences: A006485 A006486 A006487 * A006489 A006490 A006491


KEYWORD

nonn,base


AUTHOR

N. J. A. Sloane and Robert G. Wilson v


STATUS

approved



