login
The number of n-digit integers that can be written as the product of n single-digit integers. The single-digit integers need not be distinct.
0

%I #13 Sep 08 2024 08:26:20

%S 10,27,55,85,108,119,118,108,94,78,60,46,35,27,19,14,10,7,4,2,1,0,0,0,

%T 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,

%U 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0

%N The number of n-digit integers that can be written as the product of n single-digit integers. The single-digit integers need not be distinct.

%C a(21)=1 (9^21 has 21 digits). For all n>21, a(n)=0.

%H <a href="/index/Con#constant">Index entries for eventually constant sequences</a>

%e a(2) is 27 because 27 2-digit integers can be written as the product of 2 single-digit integers. Those 27 integers are: 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 35, 36, 40, 42, 45, 48, 49, 54, 56, 63, 64, 72 and 81. Note that each of the 2-digit integers 12, 16, 18, 24 and 36 can be expressed as a product of 2 single-digit integers in more than 1 way. However, each of those 2-digit integers is only counted once.

%o (Python)

%o from math import prod

%o from itertools import combinations_with_replacement as cwr

%o def a(n):

%o if n > 21: return 0

%o L, U = (n>1)*10**(n-1)-1, 10**n

%o return len(set(p for mc in cwr(range(10), n) if L < (p:=prod(mc)) < U))

%o print([a(n) for n in range(1, 22)]) # _Michael S. Branicky_, Aug 27 2024

%Y Cf. A366181.

%K nonn,base

%O 1,1

%A _Clive Tooth_, Aug 27 2024