cp's OEIS Frontend

A067734 Number of ways writing n as a product of decimal digits of some other number which has no digits equal to 1.

%I A067734 #19 Jun 15 2017 04:06:24
%S A067734 0,1,1,2,1,3,1,4,2,2,0,7,0,2,2,7,0,7,0,5,2,0,0,17,1,0,3,5,0,8,0,13,0,
%T A067734 0,2,21,0,0,0,12,0,8,0,0,5,0,0,38,1,3,0,0,0,15,0,12,0,0,0,24,0,0,5,24,
%U A067734 0,0,0,0,0,6,0,58,0,0,3,0,0,0,0,26,5,0,0,24,0,0,0,0,0,24,0,0,0,0,0,82,0
%N A067734 Number of ways writing n as a product of decimal digits of some other number which has no digits equal to 1.
%C A067734 For n=36, this was given as an exercise for children of age 14 years.
%H A067734 David A. Corneth, <a href="/A067734/b067734.txt">Table of n, a(n) for n = 1..10000</a>
%F A067734 a(A002473(n)) > 0 for n > 1. - _David A. Corneth_, Jun 14 2017
%e A067734 There are 21 other numbers with no digit 1 whose digit product equals 36: 49, 66, 94, 229, 236, 263, 292, 326, 334, 343, 362, 433, 623, 632, 922, 2233, 2323, 2332, 3223, 3232, 3322. If 1-digits were permitted then an infinite number of solutions would exist, e.g., 111114111113111113. If n has a prime divisor larger than 7, i.e., a prime divisor that is two or more digits in length, such as 11, then no solutions exist at all. The largest solution is a (decimal) number created by concatenating not-necessarily-distinct prime factors, such as 36 = 3*2*2*2. [edited by _Jon E. Schoenfield_, Jun 14 2017]
%t A067734 id1[x_] := IntegerDigits[x]; id2[x_] := DeleteCases[id1[x], 1] f[x_] := Apply[Times, IntegerDigits[x]]; k=0; Do[s=f[n]; If[Equal[s, 36]&&!Greater[Length[id1[n]], Length[id2[n]]], k=k+1; Print[{k, n}]], {n, 1, 3400}]
%o A067734 (PARI) { A067734(n) = local(v,r,i2,i3); v=vector(4,i,valuation(n,prime(i))); if(n==1||n!=prod(i=1,4,prime(i)^v[i]), return(0)); r=0; for(i6=0,min(v[1],v[2]), for(i8=0,(v[1]-i6)\3, for(i4=0,(v[1]-i6-3*i8)\2, i2=v[1]-i6-3*i8-2*i4; for(i9=0,(v[2]-i6)\2, i3=v[2]-i6-2*i9; r += (i2+i3+i4+v[3]+i6+v[4]+i8+i9)! / i2! / i3! / i4! / v[3]! / i6! / v[4]! / i8! / i9! )))); r } \\ _Max Alekseyev_, Sep 19 2009
%Y A067734 Cf. A000073, A001222, A002473, A068183-A068187, A068189-A068191.
%K A067734 base,nonn
%O A067734 1,4
%A A067734 _Labos Elemer_, Jan 28 2002