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

%I A340747 #47 Jan 22 2022 08:42:04
%S A340747 24,40,60,67,72,88,96,100,120,132,136,144,147,150,160,168,180,184,200,
%T A340747 204,216,220,227,232,240,264,267,276,280,288,300,307,312,323,328,330,
%U A340747 340,348,352,360,367,376,384,387,396,400,408,420,424
%N A340747 Numbers in array A322744 that do not have a unique decomposition into numbers of A307002.
%C A340747 For i >= 2, A322744(i, a(n)) is in the sequence.
%C A340747 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).
%C A340747 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.
%H A340747 David Lovler, <a href="/A340747/a340747.txt">The first 49 terms and their decompositions in A322744(n,k)</a>.
%e A340747 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.
%e A340747 88 is in the sequence because 88 = A322744(3,22) = A322744(4,15) and 3,4,15 and 22 are in A307002.
%e A340747 Examples of A322744(4,p) = A322744(7,q) = A322744(10,r) with p = q + r:
%e A340747 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.
%e A340747 60*3 = 180 = A322744(4,30) = A322744(7,18) = A322744(10,12) and 30 = 18 + 12.
%e A340747 60*3 + 40 = 220 = A322744(4,37) = A322744(7,22) = A322744(10,15) and 37 = 22 + 15.
%e A340747 See A340746 for more examples.
%Y A340747 Cf. A307002, A322744, A340746.
%K A340747 nonn
%O A340747 1,1
%A A340747 _David Lovler_, Jan 20 2021