A340747 Numbers in array A322744 that do not have a unique decomposition into numbers of A307002.

24, 40, 60, 67, 72, 88, 96, 100, 120, 132, 136, 144, 147, 150, 160, 168, 180, 184, 200, 204, 216, 220, 227, 232, 240, 264, 267, 276, 280, 288, 300, 307, 312, 323, 328, 330, 340, 348, 352, 360, 367, 376, 384, 387, 396, 400, 408, 420, 424

Offset: 1

Author

David Lovler, Jan 20 2021

Keywords

• nonn

Comments

For i >= 2, A322744(i, a(n)) is in the sequence.

There are numbers in array A322744 that have three decompositions of the form A322744(4,p) = A322744(7,q) = A322744(10,r). In these cases, p = q + r. p, q and r need not be in A307002. There are two situations. (a) For n > 0, 60n = A322744(4,10n) = A322744(7,6n) = A322744(10,4n); (b) For n >= 0, 60n+40 = A322744(4,10n+7) = A322744(7,6n+4) = A322744(10,4n+3).

A proof of p = q + r. q must be even because A322744(7,q) = even. p and r must be both odd or both even, otherwise there is the contradiction that p gets equated with a fraction. When p and r are odd, (3*4*p - 4)/2 = (3*7*q - q)/2 = (3*10*r - 10)/2. Solving for p in terms of q, and p in terms of r gives p = (5/3)*q + 1/3 and p = (5/2)*r - 1/2. Multiplying the latter by 2/3 and adding the two equations gives (5/3)*p = (5/3)*q + (5/3)*r, thus p = q + r. When p and r are even, (3*4*p)/2 = (3*7*q - q)/2 = (3*10*r)/2, and the same follows.

Examples

			60 = A322744(4,10). Also 60 = A322744(6,7) and 60 = A322744(2,20). These decompositions are the same but different from A322744(4,10) as follows. 6 = A322744(2,2) and 20 = A322744(2,7), making 60 = A322744(A322744(2,2), 7) and 60 = A322744(2, A322744(2,7)). Thus 60 can be written as A322744(2,2,7), a well-defined composition because A322744(n,k) is associative. 2,4,7 and 10 are in A307002, thus A322744(4,10) and A322744(2,2,7) are different decompositions of 60, so 60 is in the sequence.
88 is in the sequence because 88 = A322744(3,22) = A322744(4,15) and 3,4,15 and 22 are in A307002.
Examples of A322744(4,p) = A322744(7,q) = A322744(10,r) with p = q + r:
60*1 + 40 = 100 = A322744(4,17) = A322744(7,10) = A322744(10,7) and 17 = 10 + 7, which works by commuting one of the decompositions. Note that 60 also works this way. 60 = A322744(4,10) = A322744(7,6) = A322744(10,4) and 10 = 6 + 4.
60*3 = 180 = A322744(4,30) = A322744(7,18) = A322744(10,12) and 30 = 18 + 12.
60*3 + 40 = 220 = A322744(4,37) = A322744(7,22) = A322744(10,15) and 37 = 22 + 15.
See A340746 for more examples.
		

Links

• David Lovler, The first 49 terms and their decompositions in A322744(n,k).

Crossrefs

Cf. A307002, A322744, A340746.