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%I #5 May 21 2019 22:05:27
%S 1,1,1,2,1,3,1,3,2,3,1,5,1,3,3,4,1,5,1,5,3,3,1,7,2,3,3,5,1,7,1,5,3,3,
%T 3,8,1,3,3,7,1,7,1,5,5,3,1,9,2,5,3
%N Number of integer partitions of n with a unique consecutive subsequence summing to every positive integer from 1 to n.
%C After a(0) = 1, same as A032741(n + 1) (number of proper divisors of n + 1).
%C The Heinz numbers of these partitions are given by A325764.
%e The a(1) = 1 through a(13) = 3 partitions:
%e (1) (11) (21) (1111) (221) (111111) (2221) (3311)
%e (111) (311) (4111) (11111111)
%e (11111) (1111111)
%e .
%e (22221) (1111111111) (33311) (111111111111) (2222221)
%e (51111) (44111) (7111111)
%e (111111111) (222221) (1111111111111)
%e (611111)
%e (11111111111)
%t normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
%t Table[Length[Select[IntegerPartitions[n],normQ[Total/@Union[ReplaceList[#,{___,s__,___}:>{s}]]]&&UnsameQ@@Total/@Union[ReplaceList[#,{___,s__,___}:>{s}]]&]],{n,0,20}]
%Y Cf. A000041, A002033, A103295, A103300, A143823, A169942, A325676, A325683, A325768, A325769, A325770.
%K nonn,more
%O 0,4
%A _Gus Wiseman_, May 20 2019
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