A325857 Number of integer partitions of n such that every orderless pair of distinct parts has a different sum.

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, 55, 74, 97, 125, 165, 209, 269, 335, 428, 527, 664, 804, 1005, 1210, 1496, 1780, 2186, 2586, 3148, 3698, 4473, 5226, 6279, 7290, 8706, 10067, 11950, 13744, 16242, 18605, 21864, 24942, 29184, 33188, 38651, 43782, 50791, 57402, 66300, 74683, 86026, 96658

Offset: 0

Author

Gus Wiseman, May 31 2019

Keywords

• nonn

Examples

			The A000041(14) - a(14) = 10 partitions of 14 not satisfying the condition are:
  (6,5,2,1)
  (6,4,3,1)
  (5,4,3,2)
  (5,4,2,2,1)
  (4,4,3,2,1)
  (5,4,2,1,1,1)
  (4,3,3,2,1,1)
  (4,3,2,2,2,1)
  (4,3,2,2,1,1,1)
  (4,3,2,1,1,1,1,1)
		

Links

• Max Alekseyev, Table of n, a(n) for n = 0..80

Crossrefs

The subset case is A196723.

The maximal case is A325878.

The integer partition case is A325857.

The strict integer partition case is A325877.

Heinz numbers of the counterexamples are given by A325991.

Cf. A002033, A103300, A108917, A143823, A196724, A325853, A325855, A325858, A325859, A325862.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],UnsameQ@@Plus@@@Subsets[Union[#],{2}]&]],{n,0,30}]

Extensions

Terms a(31) onward from Max Alekseyev, Sep 23 2023