A045521 Numbers k such that k! has initial digit '2'.

2, 4, 16, 20, 23, 30, 32, 34, 39, 44, 47, 54, 58, 68, 75, 85, 86, 87, 113, 114, 115, 126, 135, 142, 148, 153, 159, 163, 168, 172, 180, 184, 188, 195, 205, 208, 211, 214, 217, 220, 223, 226, 234, 237, 242, 247, 252, 257, 262, 269, 276, 278, 287, 289, 302, 304

Offset

1,1

Comments

The asymptotic density of this sequence is log_10(3/2) = 0.176091... (see A154580) (Kunoff, 1987). - Amiram Eldar, Jul 17 2020

10^10 and 10^1000 are both in this sequence: the first 10 digits of (10^10)! and (10^1000)! are '2325796205' and '2101826996', respectively. This can be computed from the fractional part of (log(2 Pi)/2 - N)/log(10), as follows from Stirling's approximation. - M. F. Hasler, Apr 03 2026

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

Formula

A008905(a(n)) = 2. - Amiram Eldar, Jul 17 2020

Example

4 is a term since 4! = 24 has the initial digit 2.

Mathematica

Select[ Range[ 310 ], IntegerDigits[ #! ] [ [ 1 ] ] == 2 & ]

Prog

(PARI) isok(n) = digits(n!)[1] == 2; \\ Michel Marcus, Feb 07 2017
(PARI) select( {is_A045521(n)=Vec(Str(factorial(n)))[1]=="2"}, [0..399]) \\ Maybe not suitable for very large n >> 10^10. - M. F. Hasler, Apr 03 2026

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, A154580.

Sequence in context: A229238 A212202 A102545 * A338766 A242205 A076434

Adjacent sequences: A045518 A045519 A045520 * A045522 A045523 A045524

Keyword

nonn,base

Author

Jeff Burch

Extensions

Offset changed to 1 by Chai Wah Wu, Feb 07 2017

Status

approved