A348318 Number of n-digit Niven (or Harshad) numbers not containing the digit 0.

9, 14, 108, 710, 4978, 35724, 273032, 2097356, 16674554, 135091242, 1112325268, 9296413365, 77991481271, 654495034497, 5420117473932

Offset

1,1

Comments

This sequence comes from a problem, proposed by Argentina during the 39th International Mathematical Olympiad in 1998 at Taipei, Taiwan, but not used for the competition. This problem asked for a proof that, for each positive integer n, there exists an n-digit number, not containing the digit 0 and that is divisible by the sum of its digits (see links: Diophante in French and Scholes in English).

This sequence gives the number of such n-digit Niven numbers not containing the digit 0.

Links

Table of n, a(n) for n=1..15.

Diophante, Bon souvenir de Buenos-Aires.

John Scholes, 39th IMO 1998 shortlisted problems, problem N7.

Index to sequences related to Olympiads.

Example

a(2) = 14 because there are 14 integers {12, 18, 21, 24, 27, 36, 42, 45, 48, 54, 63, 72, 81, 84} in the set of Niven numbers with 2 digits and not containing the digit 0.

Mathematica

q[n_] := ! MemberQ[(d = IntegerDigits[n]), 0] && Divisible[n, Plus @@ d]; a[n_] := Module[{c = 0, k = (10^n - 1)/9}, While[k < 10^n, If[q[k], c++]; k++]; c]; Array[a, 6] (* Amiram Eldar, Oct 17 2021 *)

Prog

(Python)
from itertools import product
def a(n): return sum(1 for p in product("123456789", repeat=n) if int("".join(p))%sum(map(int, p)) == 0)
print([a(n) for n in range(1, 8)]) # Michael S. Branicky, Oct 17 2021

Crossrefs

Cf. A005349, A009994, A140866, A217973, A348150, A348316, A348317.

Sequence in context: A272009 A298754 A166654 * A370674 A131224 A073920

Adjacent sequences: A348315 A348316 A348317 * A348319 A348320 A348321

Keyword

nonn,base,more

Author

Bernard Schott, Oct 17 2021

Extensions

a(6)-a(10) from Amiram Eldar, Oct 17 2021

a(11)-a(15) from Martin Ehrenstein, Oct 22 2021

Status

approved