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%I #23 Dec 20 2024 13:09:10
%S 1,3,15,27,75,165,345,315,693,945,1365,2457,2835,4455,3465,4095,7245,
%T 15435,12285,10395,17325,27405,24255,29925,31185,48195,70875,45045,
%U 58905,61425,89775,128205,142065,121275,135135
%N a(n) is the smallest integer k such that [1,2,...,k] contains a set of n but no more disjoint consecutive subsequences, each having element sum k.
%C We observe that a(n) == 0 (mod 3) for n>1.
%e +----+-----+------------------------------------------------+
%e | n | k | n subsets whose sum of elements = k |
%e +----+-----+------------------------------------------------+
%e | 1 | 1 | {1} |
%e +----+-----+------------------------------------------------+
%e | 2 | 3 | {1,2}, {3} |
%e +----+-----+------------------------------------------------+
%e | 3 | 15 | {1,2,3,4,5}, {7,8}, {15} |
%e +----+-----+------------------------------------------------+
%e | 4 | 27 | {2,3,4,5,6,7}, {8,9,10}, {13,14}, {27} |
%e +----+-----+------------------------------------------------+
%e | 5 | 75 | {3,4,5,6,7,8,9,10,11,12}, {13,14,15,16,17}, |
%e | | | {24,25,26}, {37,38}, {75} |
%e +----+-----+------------------------------------------------+
%e | 6 | 165 | {4,5,...,18}, {25,26,...,30}, {31,32,33,34,35},|
%e | | | {54,55,56}, {82,83}, {165} |
%e +----+-----+------------------------------------------------+
%e | 7 | 345 | {4,5,...,26}, {30,31,...,39}, {55,56,...60}, |
%e | | | {67,68,...71}, {114,115,116}, {172,173},{345} |
%e +----+-----+------------------------------------------------+
%e Note: There are several solutions to reach n disjoint subsets, for example for n=3, we can choose from one of the 2 solutions: {{4,5,6},{7,8},{15}} or {{1,2,3,4,5}, {7,8}, {15}}.
%K nonn,more
%O 1,2
%A _Michel Lagneau_, Dec 05 2024
%E a(10)-a(35) from _Alois P. Heinz_, Dec 06 2024