A063525 Sum divides product: number of ordered triples of positive solutions (r,s,t) to the equation rst = n(r+s+t).

6, 15, 28, 30, 48, 45, 45, 78, 75, 54, 84, 94, 48, 105, 132, 105, 84, 99, 78, 189, 138, 60, 111, 210, 90, 132, 184, 129, 114, 153, 102, 228, 141, 105, 294, 267, 48, 132, 234, 228, 132, 159, 78, 300, 270, 96, 159, 301, 144, 231, 228, 162, 120, 297, 270, 429, 144, 72

Offset: 1

Author

Jud McCranie Aug 01 2001

Keywords

• nonn

Examples

			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.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..500
• M. J. Pelling, Problem 10745, Amer. Math. Monthly, vol. 106 (1999), p. 587.
• M. J. Pelling and F. W. Roush, The Sum Divides the Product: Problem 10745, Amer. Math. Monthly, vol. 108, (no. 7, Aug. 2001), pp. 668-669. [Gives upper bound]

Crossrefs

Cf. A063520.

Programs

Mathematica

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 *)

Extensions

More terms from David W. Wilson, Aug 01 2001