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A343207 Numbers k such that there exists a unique partition of k into positive integers x,y,z such that x+y, y+z and x+z divide x*y, y*z and x*z, respectively.

%I A343207 #20 Apr 09 2021 11:35:10
%S A343207 6,12,15,18,20,28,35,36,40,54,56,63,70,75,77,78,88,91,99,100,102,104,
%T A343207 108,114,117,130,138,143,153,154,162,170,174,175,176,182,184,186,187,
%U A343207 189,190,196,200,208,209,221,222,238,245,246,247,258,261,266,272,282,286,297
%N A343207 Numbers k such that there exists a unique partition of k into positive integers x,y,z such that x+y, y+z and x+z divide x*y, y*z and x*z, respectively.
%C A343207 A subsequence of A005279, except for terms such as 184, 261, 568, 826, 848, ..., which partition into distinct parts.
%C A343207 A variant of A343126.
%H A343207 Alois P. Heinz, <a href="/A343207/b343207.txt">Table of n, a(n) for n = 1..1000</a>
%e A343207 15 = 3+6+6 with 3+6 = 9 | 18 and 6+6 = 12 | 36.
%t A343207 sel[k_] := Select[IntegerPartitions[k, {3}], ({x, y, z} = Sort[#]; Divisible[x y, x+y] && Divisible[y z, y+z] && Divisible[x z, x+z])&];
%t A343207 Reap[For[k = 3, k <= 500, k++, sk = sel[k]; If[Length[sk] == 1, Print[k, " ", Sort[sk[[1]]]]; Sow[k]]]][[2, 1]]
%Y A343207 Cf. A005279, A010814, A343126.
%K A343207 nonn
%O A343207 1,1
%A A343207 _Jean-François Alcover_, Apr 08 2021