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

1,5

Comments

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

Links

Table of n, a(n) for n=1..70.

Formula

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

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

Example

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.

Mathematica

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

Prog

(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

Crossrefs

Cf. A080221, A138530, A356555.

Sequence in context: A259743 A247015 A241972 * A226770 A185782 A304027

Adjacent sequences: A357820 A357821 A357822 * A357824 A357825 A357826

Keyword

nonn,base

Author

Rémy Sigrist, Oct 17 2022

Status

approved