A122977 Number of sublattices of the divisor lattice of divisors of n that include n.

1, 2, 2, 4, 2, 7, 2, 8, 4, 7, 2, 21, 2, 7, 7, 16, 2, 21, 2, 21, 7, 7, 2, 58, 4, 7, 8, 21, 2, 45, 2, 32, 7, 7, 7, 84, 2, 7, 7, 58, 2, 45, 2, 21, 21, 7, 2, 152, 4, 21, 7, 21, 2, 58, 7, 58, 7, 7, 2, 200, 2, 7, 21, 64, 7, 45, 2, 21, 7, 45, 2, 293, 2, 7, 21, 21, 7, 45, 2, 152, 16, 7, 2, 200, 7, 7

Offset: 1

Author

Franklin T. Adams-Watters, Sep 21 2006

Keywords

• nonn
• nice

Comments

A divisor lattice is closed under GCD and LCM. First differences of A074986. Depends only on the prime signature of n.

Examples

			The a(6) = 7 sublattices of {1,2,3,6} that include 6 are: {6}, {1,6}, {2,6}, {3,6}, {1,2,6}, {1,3,6}, {1,2,3,6}.
		

Crossrefs

Cf. A051839, A074986, A122978, A122979, A122980, A326878.

Programs

Mathematica

okQ[dd_List] := AllTrue[Subsets[dd, {2}], MemberQ[dd, GCD @@ #] && MemberQ[dd, LCM @@ #]&];
a[n_] := Select[Rest @ Subsets[Divisors[n]], Last[#] == n && okQ[#]&] // Length;
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Aug 18 2022 *)

Formula

a(A002110(n)) = A326878(n). - Andrew Howroyd, Apr 17 2020