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

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, 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, 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

Offset

1,4

Comments

For n=36, this was given as an exercise for children of age 14 years.

Links

David A. Corneth, Table of n, a(n) for n = 1..10000

Formula

a(A002473(n)) > 0 for n > 1. - David A. Corneth, Jun 14 2017

Example

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]

Mathematica

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}]

Prog

(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

Crossrefs

Cf. A000073, A001222, A002473, A068183-A068187, A068189-A068191.

Sequence in context: A132460 A238800 A375266 * A303758 A161904 A360678

Adjacent sequences: A067731 A067732 A067733 * A067735 A067736 A067737

Keyword

base,nonn

Author

Labos Elemer, Jan 28 2002

Status

approved