cp's OEIS Frontend

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 + ...

%I A059519 #11 Jul 27 2019 14:57:51
%S A059519 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,
%T A059519 36,37,38,40,41,44,48,50,52,56,64,65,66,67,68,69,70,72,73,74,80,81,84,
%U A059519 88,96,98,100,104,112,116,128,129,130,131,132,133,134,136,137,138,139,140
%N 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 + ...
%C A059519 Partition encoding as in A029931. Complement of A059520.
%C A059519 From _Gus Wiseman_, Jul 22 2019: (Start)
%C A059519 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:
%C A059519    1: (1)
%C A059519    2: (2)
%C A059519    3: (2,1)
%C A059519    4: (3)
%C A059519    5: (3,1)
%C A059519    6: (3,2)
%C A059519    8: (4)
%C A059519    9: (4,1)
%C A059519   10: (4,2)
%C A059519   11: (4,2,1)
%C A059519   12: (4,3)
%C A059519   14: (4,3,2)
%C A059519   16: (5)
%C A059519   17: (5,1)
%C A059519   18: (5,2)
%C A059519   19: (5,2,1)
%C A059519   20: (5,3)
%C A059519 (End)
%e A059519 14=2+4+8 so Partition(14) = [2,3,4], whose sub-sums are 0,2,3,4,5,6,7 and 14.
%t A059519 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A059519 Select[Range[100],UnsameQ@@Total/@Subsets[bpe[#]]&] (* _Gus Wiseman_, Jul 22 2019 *)
%Y A059519 Cf. A000120, A029931, A048793, A059520, A070939, A108917, A272020, A275972, A299702, A326015.
%Y A059519 Other sequences classifying numbers by their binary indices: A291166 (relatively prime), A295235 (arithmetic progression), A326669 (integer average), A326675 (pairwise coprime).
%K A059519 easy,nonn
%O A059519 1,2
%A A059519 _Marc LeBrun_, Jan 19 2001