%I #11 Jun 19 2021 22:29:05

%S 1,2,3,4,4,6,7,9,13,12,17,21,28,26,49,46,74,68,113,107,176,144,255,

%T 235,375

%N Number of necklace compositions of n such that every distinct circular subsequence has a different sum.

%C A necklace composition of n is a finite sequence of positive integers summing to n that is lexicographically minimal among all of its cyclic rotations.

%C A circular subsequence is a sequence of consecutive terms where the first and last parts are also considered consecutive.

%e The a(1) = 1 through a(8) = 13 necklace compositions:

%e (1) (2) (3) (4) (5) (6) (7) (8)

%e (11) (12) (13) (14) (15) (16) (17)

%e (111) (22) (23) (24) (25) (26)

%e (1111) (11111) (33) (34) (35)

%e (222) (124) (44)

%e (111111) (142) (125)

%e (1111111) (152)

%e (2222)

%e (11111111)

%t neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];

%t subalt[q_]:=Union[ReplaceList[q,{___,s__,___}:>{s}],DeleteCases[ReplaceList[q,{t___,__,u___}:>{u,t}],{}]];

%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],neckQ[#]&&UnsameQ@@Total/@subalt[#]&]],{n,20}]

%Y Cf. A000079, A000740, A008965, A059966, A108917, A143823, A169942, A276024.

%Y Cf. A325676, A325680, A325685, A325687.

%K nonn,more

%O 1,2

%A _Gus Wiseman_, May 13 2019

%E a(21)-a(25) from _Robert Price_, Jun 19 2021