A059519 Number of partitions of n all of whose subpartitions sum to distinct values. Partition(n) = [a, b, c...] where 2n = 2^a + 2^b + 2^c + ...

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 16, 17, 18, 19, 20, 21, 24, 26, 28, 32, 33, 34, 35, 36, 37, 38, 40, 41, 44, 48, 50, 52, 56, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 80, 81, 84, 88, 96, 98, 100, 104, 112, 116, 128, 129, 130, 131, 132, 133, 134, 136, 137, 138, 139, 140

Offset: 1

Author

Marc LeBrun, Jan 19 2001

Keywords

• easy
• nonn

Comments

Partition encoding as in A029931. Complement of A059520.

From Gus Wiseman, Jul 22 2019: (Start)

These are numbers whose positions of 1's in their reversed binary expansion form a strict knapsack partition (A275972). The initial terms together with their corresponding partitions are:

1: (1)

2: (2)

3: (2,1)

4: (3)

5: (3,1)

6: (3,2)

8: (4)

9: (4,1)

10: (4,2)

11: (4,2,1)

12: (4,3)

14: (4,3,2)

16: (5)

17: (5,1)

18: (5,2)

19: (5,2,1)

20: (5,3)

(End)

Examples

			14=2+4+8 so Partition(14) = [2,3,4], whose sub-sums are 0,2,3,4,5,6,7 and 14.
		

Crossrefs

Cf. A000120, A029931, A048793, A059520, A070939, A108917, A272020, A275972, A299702, A326015.

Other sequences classifying numbers by their binary indices: A291166 (relatively prime), A295235 (arithmetic progression), A326669 (integer average), A326675 (pairwise coprime).

Programs

Mathematica

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[100],UnsameQ@@Total/@Subsets[bpe[#]]&] (* Gus Wiseman, Jul 22 2019 *)