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A126789 a(n) is the smallest number such that the product of its digits is n times the sum of its digits, or 0 if no such number exists.

%I A126789 #14 Jan 28 2019 02:24:09
%S A126789 1,36,66,88,257,268,279,448,369,459,0,666,0,578,579,678,0,1689,0,2558,
%T A126789 789,0,0,1899,13557,0,999,3477,0,2589,0,2688,0,0,13578,3489,0,0,0,
%U A126789 3588,0,2799,0,0,4569,0,0,4668,4677,5568,0,0,0,3699,0,3789,0,0,0,4599,0,0
%N A126789 a(n) is the smallest number such that the product of its digits is n times the sum of its digits, or 0 if no such number exists.
%C A126789 a(11) = 0. Proof: 11 is a prime number and the product of digits of a number in base 10 can never be a multiple of 11. - _Stefan Steinerberger_, Jun 07 2007
%C A126789 More generally, a(n) = 0 for all n which are divisible by a prime bigger than 7. This means that the sequence will almost always be 0 (with the set of exceptions having density 0). In each term the digits will be increasing (otherwise we could rearrange the digits so that they form a smaller number with the specified property). If no prime factors of n exceed 7, does this mean that a(n) is not 0? - _Stefan Steinerberger_, Jun 14 2007
%H A126789 Chai Wah Wu, <a href="/A126789/b126789.txt">Table of n, a(n) for n = 1..10000</a>
%e A126789 a(2)=36 because 3*6/(3+6) = 2 and no number smaller than 36 has this property.
%p A126789 for n from 1 to 10 do b:=proc(k) local kk: kk:=convert(k,base,10): if product(kk[j],j=1..nops(kk))=n*sum(kk[j],j=1..nops(kk)) then k else fi end: a[n]:=[seq(b(k),k=1..1000)][1]: od: seq(a[n],n=1..10); # program works only for n from 1 to 10 _Emeric Deutsch_, Mar 07 2007
%t A126789 a[1] := 1; a[n_] := Module[{}, k = 0; If[FactorInteger[n][[ -1, 1]] < 8, k = 1; While[Times @@ IntegerDigits[k] != n*Plus @@ IntegerDigits[k], k++ ]]; k]; Table[a[i], {i, 1, 80}] (* _Stefan Steinerberger_, Jun 14 2007 *)
%Y A126789 This sequence is a subsequence of A061013 (Product of digits of n) is divisible by (sum of digits of n), where 0's are not permitted.
%K A126789 base,nonn
%O A126789 1,2
%A A126789 _Tanya Khovanova_, Feb 19 2007
%E A126789 More terms from _Emeric Deutsch_, Mar 07 2007
%E A126789 More terms from _Stefan Steinerberger_, Jun 14 2007