A348150 a(n) is the smallest Niven (or Harshad) number with exactly n digits and not containing the digit 0.

1, 12, 111, 1116, 11112, 111114, 1111112, 11111112, 111111111, 1111111125, 11111111112, 111111111126, 1111111111116, 11111111111114, 111111111111114, 1111111111111122, 11111111111111112, 111111111111111132, 1111111111111111119, 11111111111111111121, 111111111111111111117

Offset: 1

Author

Bernard Schott, Oct 03 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 smallest such n-digit integer.

Examples

			111114 has 6 digits, does not contain 0 and is divisible by 1+1+1+1+1+4 = 9 (111114 = 9*12346), while 111111, 111112, 111113 are not respectively divisible by sum of their digits: 6, 7, 8; hence, a(6) = 111114.
		

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. A002275, A005349, A217973.

Programs

Mathematica

hQ[n_] := ! MemberQ[(d = IntegerDigits[n]), 0] && Divisible[n, Plus @@ d]; a[n_] := Module[{k = (10^n - 1)/9}, While[! hQ[k], k++]; k]; Array[a, 30] (* Amiram Eldar, Oct 03 2021 *)

PARI

a(n) = for(k=(10^n-1)/9, 10^n-1, if (vecmin(digits(k)) && !(k % sumdigits(k)), return (k)));  \\ Michel Marcus, Oct 03 2021

Python

def niven(n):
    s = str(n)
    return '0' not in s and n%sum(map(int, s)) == 0
def a(n):
    k = int("1"*n)
    while not niven(k): k += 1
    return k
print([a(n) for n in range(1, 22)]) # Michael S. Branicky, Oct 09 2021

Formula

a(n) = A002275(n) = R_n iff n is in A014950.

Extensions

More terms from Amiram Eldar, Oct 03 2021