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

Author

Charles R Greathouse IV, Oct 16 2012

Keywords

• nonn
• base
• easy

Comments

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

References

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

Links

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

Crossrefs

Intersection of A005349 and A052382.

A216405 is a subsequence.

Cf. A348150, A348316, A348317, A348318.

Programs

Maple

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

Mathematica

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

PARI

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

Python

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