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

1,1

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.

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.

Formula

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

Example

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.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A086627 A024118 A166902 * A275394 A236385 A335670

Adjacent sequences: A348313 A348314 A348315 * A348317 A348318 A348319

Keyword

nonn,base

Author

Bernard Schott, Oct 11 2021

Extensions

More terms from Amiram Eldar, Oct 11 2021

Status

approved