

A018799


Smallest nonnegative integer m such that m! begins with n in base 10.


13



0, 2, 9, 8, 7, 3, 6, 14, 96, 27, 22, 5, 15, 42, 25, 89, 69, 76, 63, 16, 87, 113, 54, 4, 23, 30, 205, 85, 34, 28, 62, 164, 41, 245, 17, 9, 36, 128, 11, 8, 185, 53, 351, 73, 369, 118, 12, 265, 129, 7, 21, 38, 235, 66, 46, 258, 81, 597, 279, 43, 72, 13, 559, 18, 203, 120, 311
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OFFSET

1,2


COMMENTS

Record high values are m = 0, 2, 9, 14, 96, 113, 205, 245, 351, 369, 597, ... (see A279089); these occur, respectively, at n = 1, 2, 3, 8, 9, 22, 27, 34, 43, 45, 58, ... (see A279090).  Jon E. Schoenfield, Jan 30 2017
The existence of such m for each n was proven by Maxfield in 1970. The first 999 terms of this sequence were calculated by Southard in 1983.  Amiram Eldar, Dec 18 2018


LINKS

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (terms 1..1000 from David W. Wilson).
John E. Maxfield, A Note on N!, Mathematics Magazine, Vol. 43. No. 2 (1970), pp. 6467.
Laura Southard, Investigations on Maxfield's Theorem, Pi Mu Epsilon Journal, Vol. 7, No. 8 (1983), pp. 493495, alternative link.


EXAMPLE

Since no factorial below 96! ~ 9.91*10^149 starts with 9, we have a(9) = 96. Similarly, 16 first appears as the leading digits of 89! ~ 1.65*10^136 and hence a(16) = 89.  Lekraj Beedassy, Oct 31 2010 and Robert G. Wilson v, Nov 05 2010


MATHEMATICA

f[n_] := Block[{k = 0, m}, While[ m = Max[0, Floor@ Log[10, k! ]  Floor@ Log[10, n]]; (k!  Mod[k!, 10^m])/10^m != n, k++ ]; k]; Array[f, 67] (* Robert G. Wilson v, Nov 05 2010 *)


PROG

(PARI) A018799(n)={ local( F=1, k=0 ); while( F\1!=n, F*=k++; while( F>=n+1, F/=10)); k} \\ M. F. Hasler, Feb 01 2009


CROSSREFS

Cf. A018854.
Apart from leading term, identical to A076219.
Cf. A000142, A008905, A279089, A279090.
Sequence in context: A021975 A192599 A021081 * A076219 A077601 A090930
Adjacent sequences: A018796 A018797 A018798 * A018800 A018801 A018802


KEYWORD

nonn,base


AUTHOR

David W. Wilson


STATUS

approved



