A299701 Number of distinct subset-sums of the integer partition with Heinz number n.

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 5, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 6, 3, 4, 4, 6, 2, 7, 2, 6, 4, 4, 4, 7, 2, 4, 4, 7, 2, 8, 2, 6, 6, 4, 2, 7, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 8, 2, 4, 5, 7, 4, 8, 2, 6, 4, 7, 2, 8, 2, 4, 6, 6, 4, 8, 2, 8, 5, 4, 2, 9, 4, 4, 4

Offset

1,2

Comments

An integer n is a subset-sum of an integer partition y if there exists a submultiset of y with sum n. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Position of first appearance of n appears to be A259941(n-1) = least Heinz number of a complete partition of n-1. - Gus Wiseman, Nov 16 2023

Links

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Formula

a(n) <= A000005(n) and a(n) = A000005(n) iff n is the Heinz number of a knapsack partition (A299702).

Example

The subset-sums of (5,1,1,1) are {0, 1, 2, 3, 5, 6, 7, 8} so a(88) = 8.
The subset-sums of (4,3,1) are {0, 1, 3, 4, 5, 7, 8} so a(70) = 7.

Mathematica

Table[Length[Union[Total/@Subsets[Join@@Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]], {n, 100}]

Crossrefs

Cf. A000005, A000041, A000720, A001222, A056239, A108917, A112798, A122111, A122768, A215366, A276024, A284640, A296150, A299702.

Positions of first appearances are A259941.

The triangle for this rank statistic is A365658.

The semi version is A366739, sum A366738, strict A366741.

Cf. A032302, A070933, A304792, A304793, A365543, A365920, A367095.

Sequence in context: A327527 A337454 A289849 * A286605 A035149 A074848

Adjacent sequences: A299698 A299699 A299700 * A299702 A299703 A299704

Keyword

nonn

Author

Gus Wiseman, Feb 17 2018

Extensions

Comment corrected by Gus Wiseman, Aug 09 2024

Status

approved