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%I #19 Feb 21 2022 14:55:30
%S 1,1,1,3,1,5,3,5,3,9,1,9,5,7,5,11,1,13,5,9,5,13,3,13,7,9,5,17,1,17,5,
%T 9,9,15,5,15,5,13,5,21,1,17,9,9,9,17,3,21,7,13,5,17,5,21,9,13,5,21,1,
%U 21,9,11,13,19,5,17,5,17,5,29,1,21,9,9,13,17,5,25,7,17,7
%N Number of compositions of n whose distinct consecutive subsequences have different sums, and such that these sums cover an initial interval of positive integers.
%C A composition of n is a finite sequence of positive integers summing to n.
%C Compare to the definition of perfect partitions (A002033).
%H Fausto A. C. Cariboni, <a href="/A325685/b325685.txt">Table of n, a(n) for n = 0..100</a>
%H Fausto A. C. Cariboni, <a href="/A325685/a325685.txt">All compositions that yield a(n) for n = 1..100</a>, Feb 21 2022.
%e The distinct consecutive subsequences of (3,4,1,1) together with their sums are:
%e 1: {1}
%e 2: {1,1}
%e 3: {3}
%e 4: {4}
%e 5: {4,1}
%e 6: {4,1,1}
%e 7: {3,4}
%e 8: {3,4,1}
%e 9: {3,4,1,1}
%e Because the sums are all different and cover {1...9}, it follows that (3,4,1,1) is counted under a(9).
%e The a(1) = 1 through a(9) = 9 compositions:
%e 1 11 12 1111 113 132 1114 1133 1143
%e 21 122 231 1222 3311 1332
%e 111 221 111111 2221 11111111 2331
%e 311 4111 3411
%e 11111 1111111 11115
%e 12222
%e 22221
%e 51111
%e 111111111
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Sort[Total/@Union[ReplaceList[#,{___,s__,___}:>{s}]]]==Range[n]&]],{n,0,15}]
%Y Cf. A000079, A002033, A103295, A126796, A143823, A169942, A325676, A325677, A325683, A325684.
%K nonn
%O 0,4
%A _Gus Wiseman_, May 13 2019
%E a(21)-a(25) from _Jinyuan Wang_, Jun 26 2020
%E a(21)-a(25) corrected, a(26)-a(80) from _Fausto A. C. Cariboni_, Feb 21 2022