A348316 a(n) is the largest Niven (or Harshad) number with exactly n digits and not containing the digit 0.

9, 84, 999, 9963, 99972, 999984, 9999966, 99999966, 999999999, 9999999828, 99999999898, 999999999853, 9999999999936, 99999999999783, 999999999999984, 9999999999999858, 99999999999999939, 999999999999999831, 9999999999999999951, 99999999999999999922, 999999999999999999687

Offset: 1

Author

Bernard Schott, Oct 11 2021

Keywords

• nonn
• base

Comments

This sequence is inspired by a problem, proposed by Argentina during the 39th International Mathematical Olympiad in 1998 at Taipei, Taiwan, but not used for the competition.

The problem asked for a proof that, for each positive integer n, there exists a n-digit number, not containing the digit 0 and that is divisible by the sum of its digits (see links: Diophante in French and Kalva in English).

This sequence lists the largest such n-digit integer.

Examples

			9963 has 4 digits, does not contain 0 and is divisible by 9+9+6+3 = 27 (9963 = 27*369), while there is no integer k with 9964 <= k <= 9999 that is divisible by sum of its digits, hence a(4) = 9963.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..1000
• Diophante, Bon souvenir de Buenos-Aires.
• Kalva, 39th IMO 1998 shortlisted problems, problem N7.
• Index to sequences related to Olympiads.

Crossrefs

Cf. A002283, A005349, A014950, A217973, A348150, A348317, A348318.

Programs

Mathematica

hQ[n_] := ! MemberQ[(d = IntegerDigits[n]), 0] && Divisible[n, Plus @@ d]; a[n_] := Module[{k = 10^n}, While[! hQ[k], k--]; k]; Array[a, 20] (* Amiram Eldar, Oct 11 2021 *)

Python

def a(n):
    s, k = "9"*n, int("9"*n)
    while '0' in s or k%sum(map(int, s)): k -= 1; s = str(k)
    return k
print([a(n) for n in range(1, 22)]) # Michael S. Branicky, Oct 11 2021

Formula

a(n) = A002283(n) = 10^n - 1 iff n is in A014950 (compare with A348150 formula).

Extensions

More terms from Amiram Eldar, Oct 11 2021