A045528 Numbers k such that k! has initial digit '9'.

96, 97, 98, 99, 100, 101, 102, 103, 122, 139, 166, 190, 233, 241, 261, 268, 301, 331, 366, 380, 415, 431, 445, 481, 497, 514, 521, 583, 596, 624, 629, 634, 655, 672, 690, 716, 723, 762, 789, 799, 821, 833, 861, 897

Offset: 1

Author

Jeff Burch

Keywords

• nonn
• base

Comments

The asymptotic density of this sequence is log_10(10/9) = 1 - log_10(9) = 0.045757... (A104140) (Kunoff, 1987). - Amiram Eldar, Jul 17 2020

Examples

			96 is a term since 96! = 9.916779... * 10^149 has the initial digit 9.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000
• Sharon Kunoff, N! has the first digit property, The Fibonacci Quarterly, Vol. 25, No. 4 (1987), pp. 365-367.
• Index entries for sequences related to factorial numbers

Crossrefs

For factorials with initial digit d (1 <= d <= 9) see A045509, A045510, A045511, A045516, A045517, A045518, A282021, A045519; A045520, A045521, A045522, A045523, A045524, A045525, A045526, A045527, A045528, A045529. See also A000142, A008905, A104140.

Programs

Mathematica

Select[ Range[ 900 ], IntegerDigits[ #! ] [ [1] ] == 9 & ]

PARI

isok(n) = digits(n!)[1] == 9; \\ Michel Marcus, Feb 08 2017

Formula

A008905(a(n)) = 9. - Amiram Eldar, Jul 17 2020

Extensions

More terms from Robert G. Wilson v, Jan 03 2001