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

1,2

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.

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A002275, A005349, A217973.

Sequence in context: A253091 A123933 A199691 * A077754 A199357 A293872

Adjacent sequences: A348147 A348148 A348149 * A348151 A348152 A348153

Keyword

nonn,base

Author

Bernard Schott, Oct 03 2021

Extensions

More terms from Amiram Eldar, Oct 03 2021

Status

approved