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%I #20 Mar 21 2017 10:34:04
%S 1,3,5,7,8,11,12,14,16,18,19,22,23,25,27,29,30,33,34,36,38,40,41,44,
%T 45,47,49,51,52,55,56,58,60,62,63,66,67,69,71,73,74,77,78,80,82,84,85,
%U 88,89,91,93,95,96,99,100,102,104,106,107,110,111,113,115,117,118,121,122
%N Number of positive integer triples (x,y,z), with x<=y<=z<=n, such that each of x,y and z divides the sum of the other two.
%C The following conjecture is probably not very difficult: Conjecture. The sequence (A106253) of differences of this sequence is periodic with period 6.
%C That the difference sequence in the above conjecture is periodic follows from a formula in the Formula and Mathematica sections; see A211701 for a discussion. [_Clark Kimberling_, Apr 20 2012]
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (0, 1, 1, 0, -1).
%F a(n) = n + floor(n/2) + floor(n/3). [_Clark Kimberling_, Apr 20 2012]
%e (1,1,1), (1,1,2), (1,2,3), (2,2,2) and (3,3,3) are the triples that have the desired property for n=3, so a(3)=5.
%t f[n_, m_] := Sum[Floor[n/k], {k, 1, m}]; t = Table[f[n, 3], {n, 1, 90}] (* _Clark Kimberling_, Apr 20 2012 *)
%t LinearRecurrence[{0,1,1,0,-1},{1,3,5,7,8},67] (* _Ray Chandler_, Aug 01 2015 *)
%Y Cf. A106253.
%K nonn,easy
%O 1,2
%A _John W. Layman_, Apr 27 2005