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A055471
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Divisible by the product of its nonzero digits.
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9
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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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.
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MATHEMATICA
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Select[Range[5000], IntegerQ[ #/(Times @@ Select[IntegerDigits[ # ], # > 0 &])] &] (* Alonso del Arte, Aug 04 2004 *)
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PROG
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(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
(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
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CROSSREFS
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Superset of A007602.
Cf. A007088.
Sequence in context: A246088 A071204 A002796 * A278328 A066254 A167904
Adjacent sequences: A055468 A055469 A055470 * A055472 A055473 A055474
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KEYWORD
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nonn,base
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AUTHOR
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Robert G. Wilson v, Jul 05 2000
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EXTENSIONS
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Corrected by Patrick De Geest, Aug 15 2000
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STATUS
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approved
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