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

1,2

Comments

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

Links

Table of n, a(n) for n=1..86.

Formula

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

Example

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}.

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 *)

Crossrefs

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

Sequence in context: A057767 A207329 A348219 * A275870 A321721 A359102

Adjacent sequences: A122974 A122975 A122976 * A122978 A122979 A122980

Keyword

nonn,nice

Author

Franklin T. Adams-Watters, Sep 21 2006

Status

approved