A055471 Divisible by the product of its nonzero digits.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 20, 24, 30, 36, 40, 50, 60, 70, 80, 90, 100, 101, 102, 104, 105, 110, 111, 112, 115, 120, 128, 132, 135, 140, 144, 150, 175, 200, 208, 210, 212, 216, 220, 224, 240, 250, 300, 306, 312, 315, 360, 384, 400, 432, 480, 500

Offset: 1

Author

Robert G. Wilson v, Jul 05 2000

Keywords

• nonn
• base

Comments

If n is the term then 10n also is. - Zak Seidov, Jun 09 2013

De Koninck and Luca showed that the number of terms of this sequence below x is at least x^0.495 but at most x^0.901 for sufficiently large x. - Tomohiro Yamada, Nov 18 2017

This sequence begins with a run of 12 consecutive terms, from 1 to 12. The maximal length of a run of consecutive integer terms is 13. The smallest example of such a run begins with 1111011111000 and ends with 1111011111012 (Diophante link). - Bernard Schott, Apr 26 2019

These numbers are called "nombres prodigieux" on the French site Diophante. - Bernard Schott, Apr 26 2019

Links

• Marius A. Burtea, Table of n, a(n) for n = 1..11442 (terms 1..1000 from Zak Seidov)
• Jean-Marie De Koninck and Florian Luca, Positive integers divisible by the product of their nonzero digits, Port. Math. 64 (2007) 75-85. (This proof for upper bounds contains an error. See the paper below)
• Jean-Marie De Koninck and Florian Luca, Corrigendum to "Positive integers divisible by the product of their nonzero digits", Portugaliae Math. 64 (2007), 1: 75-85, Port. Math. 74 (2017), 169-170.
• Diophante, A365, les nombres prodigieux, July 2016.
• Michael Gohn, Joshua Harrington, Sophia Lebiere, Hani Samamah, Kyla Shappell, and Tony W. H. Wong, Arithmetic Progressions of b-Prodigious Numbers, J. Int. Seq., Vol. 25 (2022), Article 22.8.7.

Crossrefs

Superset of A007602.

Cf. A007088.

Programs

MATLAB

m=1;
for n=1:1000
    v=dec2base(n,10)-'0';
    v = v(v~=0);
    if mod(n,prod(v))==0
        sol(m)=n;
        m=m+1;
    end
end
sol % Marius A. Burtea, May 07 2019

Magma

m:=1;sol:=[];
for n in [1..1000] do
      v:=Intseq(n,10);
       while &*v eq 0 do; Exclude(~v, 0); end while;
     if n mod &*(v) eq 0  then ; sol[m]:=n; m:=m+1; end if;
end for;
sol // Marius A. Burtea, May 07 2019

Mathematica

Select[Range[5000], IntegerQ[ #/(Times @@ Select[IntegerDigits[ # ], # > 0 &])] &] (* Alonso del Arte, Aug 04 2004 *)

Python

from math import prod
def ok(n): return n > 0 and n%prod([int(d) for d in str(n) if d!='0']) == 0
print(list(filter(ok, range(501)))) # Michael S. Branicky, Jul 27 2021

Extensions

Corrected by Patrick De Geest, Aug 15 2000