A006488 Numbers n such that n! has a square number of digits.

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

Offset: 1

Author

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

Keywords

• nonn
• base

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

From Bernard Schott, Jan 04 2020: (Start)

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.

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.

When m < 105 and m! has q^2 digits (m <= 81), then q <= 11 and the number of trailing zeros is <= 2*q - 3.

When m > 105 and m! has q^2 digits (m >= 132), then q >= 15 and the number of trailing zeros is >= 2*q + 2.

Hence, only 105! presents such a tree printout.

1

081

39675

8240290

900504101

30580032964

9720646107774

902579144176636

57322653190990515

3326984536526808240

339776398934872029657

99387290781343681609728

0000000000000000000000000

(End)

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

Crossrefs

Cf. A000142, A027868 (trailing zeros), A034886 (number of digits), A056851.

Programs

Magma

[k:k in [0..5000]| IsSquare(#Intseq(Factorial(k)))]; // Marius A. Burtea, Jan 04 2020

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 *)

PARI

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