A327466 Number of nonempty subsets of [1..n] which are geometric progressions with rational ratio and are locally maximal.

1, 1, 3, 4, 8, 13, 19, 23, 27, 36, 46, 55, 67, 80, 94, 103, 119, 132, 150, 167, 187, 208, 230, 250, 266, 291, 311, 336, 364, 393, 423, 447, 479, 512, 546

Offset

1,3

Comments

"Locally maximal" subsets are those subsets in geometrical progression that cannot be extended to a larger subset of [1..n] in geometric progression. [Comment made precise by Giovanni Resta, Sep 30 2019.]

One might have expected that the GP would be required to have an integer ratio, but in fact we allow rational ratios. The GPs can be assumed to be strictly increasing. - N. J. A. Sloane, Oct 03 2019

Links

Table of n, a(n) for n=1..35.

Example

Illustrations of some initial terms:
n=3: (12),(13),(23).
n=4: (124),(13),(23),(34).
n=8: (1248), plus all 28 pairs (ij) from [1..8] except the six subsets of (1248), so a(8) = 1 + 28 - 6 = 23.

Mathematica

a[1] = 1; a[n_] := Block[{t = Select[ Subsets[ Range[n], {2, Ceiling[ Log2[n + 1]]}], Length@ Union[ Rest[#]/ Most[#]] == 1 &], i = 2}, t = Reverse@ SortBy[t, Length]; i=2; While[i <= Length[t], If[ AnyTrue[ Take[t, i-1], SubsetQ[#, t[[i]]] &], t = Delete[t, i]; i=2; Continue[], i++]]; Length@ t]; Array[a, 16] (* Giovanni Resta, Sep 30 2019 *)

Crossrefs

See A327469 for GPs of length > 2.

Cf. A309095.

Sequence in context: A275989 A343562 A033955 * A049720 A078172 A022308

Adjacent sequences: A327463 A327464 A327465 * A327467 A327468 A327469

Keyword

nonn,more

Author

N. J. A. Sloane, Sep 29 2019

Extensions

a(9)-a(35) from Giovanni Resta, Sep 30 2019

Status

approved