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.

1, 36, 66, 88, 257, 268, 279, 448, 369, 459, 0, 666, 0, 578, 579, 678, 0, 1689, 0, 2558, 789, 0, 0, 1899, 13557, 0, 999, 3477, 0, 2589, 0, 2688, 0, 0, 13578, 3489, 0, 0, 0, 3588, 0, 2799, 0, 0, 4569, 0, 0, 4668, 4677, 5568, 0, 0, 0, 3699, 0, 3789, 0, 0, 0, 4599, 0, 0

Offset: 1

Author

Tanya Khovanova, Feb 19 2007

Keywords

• base
• nonn

Comments

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

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

Examples

			a(2)=36 because 3*6/(3+6) = 2 and no number smaller than 36 has this property.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000

Crossrefs

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.

Programs

Maple

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

Mathematica

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 *)

Extensions

More terms from Emeric Deutsch, Mar 07 2007

More terms from Stefan Steinerberger, Jun 14 2007