A305565 Regular triangle where T(n,k) is the number of finite sets of positive integers with least common multiple n and greatest common divisor k.

1, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 0, 0, 0, 1, 7, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 4, 2, 0, 1, 0, 0, 0, 1, 2, 0, 1, 0, 0, 0, 0, 0, 1, 7, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 32, 7, 2, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Gus Wiseman, Jun 05 2018

Keywords

• nonn
• tabl

Examples

			The T(12,2) = 7 sets are {2,12}, {4,6}, {2,4,6}, {2,4,12}, {2,6,12}, {4,6,12}, {2,4,6,12}.
Triangle begins:
   1
   1  1
   1  0  1
   2  1  0  1
   1  0  0  0  1
   7  1  1  0  0  1
   1  0  0  0  0  0  1
   4  2  0  1  0  0  0  1
   2  0  1  0  0  0  0  0  1
   7  1  0  0  1  0  0  0  0  1
   1  0  0  0  0  0  0  0  0  0  1
  32  7  2  1  0  1  0  0  0  0  0  1
		

Crossrefs

Cf. A076078, A181819, A281116, A285572, A290103, A304818, A305563, A305564, A305566, A305567.

Programs

Mathematica

Table[Length[Select[Subsets[Divisors[n]],And[GCD@@#==k,LCM@@#==n]&]],{n,20},{k,n}]

Formula

If k divides n then T(n,k) = T(n/k,1) = A305564(n/k); otherwise T(n,k) = 0.