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%I #14 Feb 25 2021 02:27:04
%S 1,1,2,3,4,6,7,11,12,17,19,29,28,41,42,62,61,88,87,123,121,168,164,
%T 234,225,306,306,411,401,527,533,700,689,894,885,1163,1150,1452,1469,
%U 1866,1835,2333,2346,2913,2913,3638,3619,4511,4537,5497,5576,6859,6827,8263
%N Number of integer partitions of n whose distinct consecutive subsequences have different sums.
%C For example (3,3,1,1) is counted under a(8) because it has distinct consecutive subsequences (), (1), (1,1), (3), (3,1), (3,1,1), (3,3), (3,3,1), (3,3,1,1), all of which have different sums.
%C The Heinz numbers of these partitions are given by A325778.
%H Fausto A. C. Cariboni, <a href="/A325769/b325769.txt">Table of n, a(n) for n = 0..300</a>
%e The a(1) = 1 through a(8) = 12 partitions:
%e (1) (2) (3) (4) (5) (6) (7) (8)
%e (11) (21) (22) (32) (33) (43) (44)
%e (111) (31) (41) (42) (52) (53)
%e (1111) (221) (51) (61) (62)
%e (311) (222) (322) (71)
%e (11111) (411) (331) (332)
%e (111111) (421) (521)
%e (511) (611)
%e (2221) (2222)
%e (4111) (3311)
%e (1111111) (5111)
%e (11111111)
%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@Total/@Union[ReplaceList[#,{___,s__,___}:>{s}]]&]],{n,0,30}]
%Y Cf. A000041, A002033, A143823, A169942, A325676, A325677, A325683, A325768, A325770, A325778.
%K nonn
%O 0,3
%A _Gus Wiseman_, May 21 2019
%E a(41)-a(53) from _Fausto A. C. Cariboni_, Feb 24 2021
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