A340746 Numbers in array A322744 that do not have a unique decomposition into numbers of A307002 and are not equal to A322744(n,k), n > 1, k in the sequence.
24, 40, 60, 67, 88, 100, 132, 136, 147, 150, 184, 204, 220, 227, 232, 276, 307, 323, 328, 330, 340, 348, 367, 376, 387, 424, 460, 472, 484, 492, 499, 510, 547, 550, 564, 567, 568, 580, 627, 636, 664, 675, 690, 707, 708, 712, 726, 748, 767
Offset: 1
Keywords
Examples
24 = A322744(4,4). Also 24 = A322744(6,3) and 24 = A322744(2,8). These two decompositions are the same but they differ from A322744(4,4) as follows. 6 = A322744(2,2) and 8 = A322744(2,3), making 24 = A322744(A322744(2,2), 3) and 24 = A322744(2, A322744(2,3)). Thus 24 can be written as A322744(2,2,3), a well-defined composition because A322744(n,k) is associative. 2,3 and 4 are in A307002, thus A322744(4,4) and A322744(2,2,3) are different decompositions of 24, so 24 is in the sequence. 40 is in the sequence because 40 = A322744(3,10) = A322744(4,7) and 3,4,7 and 10 are in A307002. 67 is in the sequence because 67 = A322744(3,17) = A322744(7,7) and 3,7 and 17 are in A307002. 220 has three decompositions. 220 = A322744(4,37) = A322744(7,22) = A322744(10,15) and 4,7,10,15,22 and 37 are in A307002. 72 = A322744(2,2,2,3) = A322744(2,4,4) is not in the sequence because 72 = A322744(2,24) and 24 is in the sequence.
Programs
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Mathematica
T[n_,k_]:=T[n,k]=(3*n*k-If[OddQ@n,If[OddQ@k,n+k-1,k],If[OddQ@k,n,0]])/2;F[d_]:=If[(q=Union[Sort/@(Position[Table[T[n,k],{n,2,Ceiling[d/3]},{k,2,Ceiling[d/3]}],d]+1)])=={},{{d}},q];FC[x_]:=FixedPoint[Union[Sort/@Flatten[Flatten/@Tuples[#]&/@((F/@#&/@#)&[#]),1]]&,F[x]];list={};Do[If[Length@FC@i>1&&ContainsNone[list,Flatten@F@i],AppendTo[list,i]],{i,500}];list (* Giorgos Kalogeropoulos, Apr 11 2021 *)
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