A348318 -id:A348318 - cp's OEIS frontend

A217973 Niven (or Harshad) numbers not containing the digit 0.

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 18, 21, 24, 27, 36, 42, 45, 48, 54, 63, 72, 81, 84, 111, 112, 114, 117, 126, 132, 133, 135, 144, 152, 153, 156, 162, 171, 192, 195, 198, 216, 222, 224, 225, 228, 234, 243, 247, 252, 261, 264, 266, 285, 288, 312, 315, 322, 324, 333, 336

Offset: 1

Charles R Greathouse IV, Oct 16 2012

Andreescu & Andrica prove that this sequence is infinite.

For each positive integer n, there exists a n-digit Niven (or Harshad) number not containing the digit 0 (see A348318 for more explanations and links). - Bernard Schott, Oct 20 2021

Titu Andreescu and Dorin Andrica, Number Theory, Structures, Examples, and Problems, Problem 5.2.3 on pages 109-110.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Intersection of A005349 and A052382.
A216405 is a subsequence.
Cf. A348150, A348316, A348317, A348318.

filter:= proc(n) local L;
L:= convert(n,base,10);
not has(L,0) and n mod convert(L,`+`) = 0
end proc:
select(filter, [$1..1000]); # Robert Israel, Apr 01 2016

Select[Range[400], IntegerQ[ #/(Plus @@ IntegerDigits[#])] && DigitCount[#, 10, 0] == 0 &]  (* Alonso del Arte, Oct 16 2012 *)

is(n)=vecsort(digits(n))[1]&&n%sumdigits(n)==0

def ok(n): s = str(n); return '0' not in s and n%sum(map(int, s)) == 0
print([k for k in range(337) if ok(k)]) # Michael S. Branicky, Oct 20 2021

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

Original entry on oeis.org

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

Offset: 1

Bernard Schott, Oct 11 2021

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.

			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.

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.

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

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 *)

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

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

More terms from Amiram Eldar, Oct 11 2021

cp's OEIS Frontend

A217973 Niven (or Harshad) numbers not containing the digit 0.

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