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%I #14 Jun 06 2019 08:56:10
%S 6,15,28,30,48,45,45,78,75,54,84,94,48,105,132,105,84,99,78,189,138,
%T 60,111,210,90,132,184,129,114,153,102,228,141,105,294,267,48,132,234,
%U 228,132,159,78,300,270,96,159,301,144,231,228,162,120,297,270,429,144,72
%N Sum divides product: number of ordered triples of positive solutions (r,s,t) to the equation rst = n(r+s+t).
%H Giovanni Resta, <a href="/A063525/b063525.txt">Table of n, a(n) for n = 1..500</a>
%H M. J. Pelling, <a href="https://www.jstor.org/stable/2589479">Problem 10745</a>, Amer. Math. Monthly, vol. 106 (1999), p. 587.
%H M. J. Pelling and F. W. Roush, <a href="https://www.jstor.org/stable/2695283">The Sum Divides the Product: Problem 10745</a>, Amer. Math. Monthly, vol. 108, (no. 7, Aug. 2001), pp. 668-669. [Gives upper bound]
%e The ordered solutions (r,s,t) of rst = 3(r+s+t) are (1,4,15), (1,5,9), (1,6,7), (2,2,12), (2,3,5), (3,3,3) for a total of 28 permuted solutions, hence a(3) = 28.
%t np[{x_,y_,z_}] := If[x==y==z, 1, If[x==y || y==z, 3, 6]]; f[n_, t_] := Block[{s, r, sol = Reduce[r s t == n (r + s + t) && r >= s >= t , {r, s}, Integers]}, If[sol === False, 0, Total[np /@ ({r, s, t} /. List@ ToRules@ sol)]]]; a[n_] := Sum[f[n, t], {t, Sqrt[3 n]}]; Array[a, 58] (* _Giovanni Resta_, Jun 06 2019 *)
%Y Cf. A063520.
%K nonn
%O 1,1
%A _Jud McCranie_ Aug 01 2001
%E More terms from _David W. Wilson_, Aug 01 2001
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