A357823 a(n) is the number of bases > 1 where n is not divisible by the sum of its digits.

0, 0, 1, 0, 3, 0, 5, 1, 4, 3, 9, 1, 11, 9, 7, 5, 15, 5, 17, 7, 11, 17, 21, 5, 18, 20, 17, 14, 27, 12, 29, 16, 24, 28, 24, 13, 35, 33, 31, 17, 39, 22, 41, 33, 26, 41, 45, 18, 42, 34, 42, 38, 51, 33, 45, 35, 48, 53, 57, 26, 59, 57, 44, 41, 52, 43, 65, 56, 60, 48

Offset: 1

Author

Rémy Sigrist, Oct 17 2022

Keywords

• nonn
• base

Comments

The sequence is well defined as the sum of digits of n equals n (and hence divides n) in any base > n.

Examples

			For n = 10, we have:
       b  sum of digits  divisible?
    ----  -------------  ----------
       2              2  Yes
       3              2  Yes
       4              4  No
       5              2  Yes
       6              5  Yes
       7              4  No
       8              3  No
       9              2  Yes
      10              1  Yes
    >=11             10  Yes
so a(n) = #{ 4, 7, 8 } = 3.
		

Crossrefs

Cf. A080221, A138530, A356555.

Programs

Mathematica

NivenQ[n_, b_] := Divisible[n, Total @ IntegerDigits[n, b]]; a[n_] := Sum[Boole @ !NivenQ[n, b], {b, 2, n}]; Array[a, 70]

PARI

a(n) = sum(b=2, n, n%sumdigits(n,b)!=0)

Python

from sympy.ntheory.factor_ import digits
def A357823(n): return sum(1 for b in range(2,n) if n%sum(digits(n,b)[1:])) # Chai Wah Wu, Oct 19 2022

Formula

a(n) = n - A080221(n).

a(p) = p - 2 for any prime number p.