A244912 Sum of leading digits in representations of n in bases 2,3,...,n.

1, 2, 3, 4, 6, 7, 9, 9, 11, 12, 15, 16, 18, 20, 20, 21, 25, 26, 29, 31, 33, 34, 38, 36, 38, 39, 42, 43, 47, 48, 52, 54, 56, 58, 58, 59, 61, 63, 67, 68, 72, 73, 76, 79, 81, 82, 88, 84, 88, 90, 93, 94, 99, 101, 105, 107, 109, 110, 116, 117, 119, 122, 117, 119, 123

Offset: 2

Author

Alex Ratushnyak, Jul 08 2014

Keywords

• nonn
• base

Examples

			8 in bases 2...8 is:
  1000 (base 2)
  22 (base 3)
  20 (base 4)
  13 (base 5)
  12 (base 6)
  11 (base 7)
  10 (base 8)
The sum of first digits is 1+2+2+1+1+1+1 = 9, so a(8)=9.
		

Links

• Jens Kruse Andersen, Table of n, a(n) for n = 2..1000

Crossrefs

Cf. A004125 (sum of last digits), A043306 (sum of all digits).

Programs

Mathematica

f[n_] := Sum[ IntegerDigits[n, k][[1]], {k, 2, n}]; Array[f, 70, 2] (* Robert G. Wilson v, Aug 02 2014 *)

PARI

a(n) = sum(i=2, n, digits(n, i)[1]); \\ Michel Marcus, Jul 17 2014

Python

import math
def modlg(a, b):
    return a // b**int(math.log(a, b))
for n in range(2,77):
    s=0
    for k in range(2,n+1):
        s += modlg(n,k)
    print(s, end=', ')

Formula

a(n) = Sum_{k=2..n} floor(n/f(n,k)), with f(n,k) = k^floor(log_k(n)). - Ridouane Oudra, Apr 26 2025