A364613 a(n) = number of partitions of n whose sum multiset is free of duplicates; see Comments.

1, 1, 2, 2, 3, 3, 5, 5, 7, 8, 10, 12, 15, 18, 20, 26, 29, 36, 38, 50, 53, 67, 69, 89, 95, 115, 122, 151, 161, 195, 201, 247, 266, 312, 330, 386, 419, 487, 520, 600, 641, 742, 793, 901, 979, 1088, 1186, 1331, 1454, 1605, 1730, 1925, 2102, 2311, 2525, 2741, 3001

Offset: 0

Author

Clark Kimberling, Sep 17 2023

Keywords

• nonn

Comments

If M is a multiset of real numbers, then the sum multiset of M consists of the sums of pairs of distinct numbers in M. For example, the sum multiset of (1,2,4,5) is {3,5,6,6,7,9}.

Examples

			The partitions of 8 are [8], [7,1], [6,2], [6,1,1], [5,3], [5,2,1], [5,1,1,1], [4,4], [4,3,1], [4,2,2], [4,2,1,1], [4,1,1,1,1], [3,3,2], [3,3,1,1], [3,2,2,1], [3,2,1,1,1], [3,1,1,1,1,1], [2,2,2,2], [2,2,2,1,1], [2,2,1,1,1,1], [2,1,1,1,1,1,1], [1,1,1,1,1,1,1,1]. The 7 partitions whose sum multiset is duplicate-free are [8], [7,1], [6,2], [5,3], [5,2,1], [4,4], [4,3,1].
		

Crossrefs

Cf. A000041, A236912, A325877, A363994.

Programs

Mathematica

s[n_, k_] := s[n, k] = Subsets[IntegerPartitions[n][[k]], {2}];
g[n_, k_] := g[n, k] = DuplicateFreeQ[Map[Total, s[n, k]]];
t[n_] := Table[g[n, k], {k, 1, PartitionsP[n]}];
a[n_] := Count[t[n], True]
Table[a[n], {n, 1, 40}]

Formula

a(n) = A325877(n) - (1 - n mod 2) for n > 0. - Andrew Howroyd, Sep 17 2023

Extensions

More terms from Alois P. Heinz, Sep 17 2023