A216377 The most significant digit in base n representation of n!.

1, 2, 1, 4, 3, 2, 1, 6, 3, 2, 1, 7, 4, 2, 1, 10, 5, 2, 1, 15, 8, 4, 2, 1, 13, 6, 3, 1, 25, 12, 6, 3, 1, 25, 12, 6, 3, 1, 28, 13, 6, 3, 1, 33, 16, 7, 3, 1, 41, 20, 9, 4, 2, 1, 26, 12, 6, 2, 1, 38, 18, 8, 3, 1, 57, 27, 12, 5, 2, 1, 43, 20, 9, 4, 2, 72, 33, 15, 7, 3, 1

Offset: 2

Author

Alex Ratushnyak, Sep 06 2012

Keywords

• nonn
• base
• look

Comments

a(n) < n, by definition.

Numbers n such that a(n)=1: 2, 4, 8, 12, 16, 20, 25, 29, 34, 39, 44, 49, 55, 60, 65, 71, 82, 88, 94, 105, 111, 117, 123, 136, ... (see A221707).

Numbers n such that a(n) > a(k) for k < n: 2, 3, 5, 9, 13, 17, 21, 30, 40, 45, 50, 66, 77, 100, 118, 124, 130, 155, 161, 226, 246, 273, 371, 378, 385, 421, 450, 472, 509, 584, 599, 637, 660, 683, 745, 784, 855, 983, 991, 999, ... (see A221708).

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A216021, A215894, A221707, A221708.

Cf. also to scatter plots of A265891 and A265894.

Programs

Maple

a:= n-> iquo(n!, n^ilog[n](n!)):
seq(a(n), n=2..100);  # Alois P. Heinz, Sep 06 2012

Mathematica

Table[IntegerDigits[n!, n][[1]], {n, 2, 100}] (* T. D. Noe, Sep 06 2012 *)

Python

import math
def modlg(a,b):
    return a // b**int(math.log(a,b))
for n in range(2,88):
    print(modlg(math.factorial(n), n), end=', ')

Formula

a(n) = modlg(n!, n), where modlg is the function defined in A215894: modlg(A,B) = floor(A / B^floor(logB(A))), logB is the logarithm base B.