A018799 Smallest nonnegative integer m such that m! begins with n in base 10.

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

Offset: 1

Author

David W. Wilson

Keywords

• nonn
• base

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

Examples

			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
		

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. 64-67.
• Laura Southard, Investigations on Maxfield's Theorem, Pi Mu Epsilon Journal, Vol. 7, No. 8 (1983), pp. 493-495, alternative link.

Crossrefs

Cf. A018854.

Apart from leading term, identical to A076219.

Cf. A000142, A008905, A279089, A279090.

Programs

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 *)

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