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

1,7

Links

Table of n, a(n) for n=1..87.

Formula

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

Example

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

Mathematica

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

Crossrefs

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

Sequence in context: A290260 A304273 A153659 * A300060 A300056 A321758

Adjacent sequences: A305562 A305563 A305564 * A305566 A305567 A305568

Keyword

nonn,tabl

Author

Gus Wiseman, Jun 05 2018

Status

approved