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A038186 Numbers divisible by the sum and product of their digits.

%I A038186 #47 Feb 16 2024 10:10:13
%S A038186 1,2,3,4,5,6,7,8,9,12,24,36,111,112,132,135,144,216,224,312,315,432,
%T A038186 612,624,735,1116,1212,1296,1332,1344,1416,2112,2232,2916,3132,3168,
%U A038186 3276,3312,4112,4224,6624,6912,8112,9612,11112,11115,11133,11172,11232
%N A038186 Numbers divisible by the sum and product of their digits.
%C A038186 The property "numbers divisible by the sum and product of their digits" leads to the Diophantine equation t*x1*x2*...*xr=s*(x1+x2+...+xr), where t and s are divisors of n; xi is from [1...9]. This corresponds to some arithmetic problems in geometry, see Sándor, 2002. - _Ctibor O. Zizka_, Mar 04 2008
%H A038186 David A. Corneth, <a href="/A038186/b038186.txt">Table of n, a(n) for n = 1..10352</a> (first 1000 terms from T. D. Noe)
%H A038186 József Sándor, <a href="http://www.gallup.unm.edu/~smarandache/JozsefSandor2.pdf">Geometric Theorems, Diophantine Equations and Arithmetic Functions</a>, American Research Press, Rehoboth, 2002.
%F A038186 A188641(a(n)) * A188642(a(n)) = 1. - _Reinhard Zumkeller_, Apr 07 2011
%t A038186 dspQ[n_]:=Module[{idn=IntegerDigits[n],t},t=Times@@idn;t!=0 && Divisible[n,Total[idn]] && Divisible[n,t]]; Select[Range[11500],dspQ] (* _Harvey P. Dale_, Jul 11 2011 *)
%o A038186 (Haskell)
%o A038186 import Data.List (elemIndices)
%o A038186 a038186 n = a038186_list !! (n-1)
%o A038186 a038186_list = map succ $ elemIndices 1
%o A038186                $ zipWith (*) (map a188641 [1..]) (map a188642 [1..])
%o A038186 -- _Reinhard Zumkeller_, Apr 07 2011
%o A038186 (PARI) for(n=1,10^4,d=digits(n);s=sumdigits(n);p=prod(i=1,#d,d[i]);if(p&&!(n%s+n%p),print1(n,", "))) \\ _Derek Orr_, Apr 29 2015
%o A038186 (Python)
%o A038186 from math import prod
%o A038186 def sd(n): return sum(map(int, str(n)))
%o A038186 def pd(n): return prod(map(int, str(n)))
%o A038186 def ok(n): return n%sd(n) == 0 and pd(n) and n%pd(n) == 0
%o A038186 def aupto(limit): return [m for m in range(1, limit+1) if ok(m)]
%o A038186 print(aupto(11233)) # _Michael S. Branicky_, Jan 28 2021
%Y A038186 Intersection of A005349 and A007602. - _Reinhard Zumkeller_, Apr 07 2011
%Y A038186 Cf. A188641, A188642.
%K A038186 nonn,base,nice,look
%O A038186 1,2
%A A038186 _Felice Russo_
%E A038186 More terms from _Patrick De Geest_, Jun 15 1999