A348317 a(n) = A348150(n) - A002275(n) where A002275(n) = R_n is the repunit with n times digit 1.

0, 1, 0, 5, 1, 3, 1, 1, 0, 14, 1, 15, 5, 3, 3, 11, 1, 21, 8, 10, 6, 5, 1, 3, 2, 12, 0, 17, 25, 14, 5, 13, 6, 74, 1, 54, 41, 12, 8, 14, 4, 105, 41, 55, 63, 33, 25, 13, 5, 103, 3, 33, 40, 63, 3, 52, 15, 23, 40, 21, 20, 10, 21, 11, 25, 33, 41, 47, 45, 14, 1, 171

Offset: 1

Author

Bernard Schott, Oct 12 2021

Keywords

• nonn
• base

Comments

a(n) measures the gap between the smallest n-digit number not containing the digit 0 and the smallest n-digit Niven number not containing the digit 0.

For more informations and links, see A348150.

a(n) = 0 iff n is in A014950.

Examples

			A348150(4) = 1116 since it is the smallest 4-digit integer not containing the digit 0 that is divisible by the sum of its digits:1116 = (1+1+1+6) * 124;  A002275(4) = R_4 = 1111, hence a(4) = 1116 - 1111 = 5.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002275, A005349, A014950, A217973, A348150.

Programs

Mathematica

hQ[n_] := ! MemberQ[(d = IntegerDigits[n]), 0] && Divisible[n, Plus @@ d]; a[n_] := Module[{m= (10^n - 1)/9,k=0}, While[! hQ[m+k], k++]; k]; Array[a, 30] (* Amiram Eldar, Oct 13 2021 *)

PARI

a(n) = my(r=(10^n-1)/9); for(k=r, 10^n-1, if (vecmin(digits(k)) && !(k % sumdigits(k)), return (k-r))); \\ Michel Marcus, Oct 13 2021

Python

def a(n):
    s, k = "1"*n, int("1"*n)
    while '0' in s or k%sum(map(int, s)): k += 1; s = str(k)
    return k - int("1"*n)
print([a(n) for n in range(1, 73)]) # Michael S. Branicky, Oct 12 2021

Formula

a(n) = A348150(n) - A002275(n).

Extensions

a(23) and beyond from Michael S. Branicky, Oct 12 2021