A063520 Sum divides product: number of solutions (r,s,t), r>=s>=t>0, to the equation rst = n(r+s+t).

1, 3, 6, 5, 8, 8, 8, 14, 13, 9, 14, 17, 8, 18, 23, 18, 14, 17, 13, 33, 23, 10, 19, 36, 15, 22, 32, 22, 19, 26, 17, 39, 24, 18, 50, 45, 8, 22, 39, 38, 22, 27, 13, 50, 45, 16, 27, 52, 24, 39, 38, 27, 20, 50, 45, 72, 24, 12, 31, 58, 15, 28, 69, 45, 49, 39, 12, 52, 40, 33, 33, 66, 12, 33, 64

Offset: 1

Author

Jud McCranie and Vladeta Jovovic, Aug 01 2001

Keywords

• nonn

Comments

Number of solutions (r,s) in positive integers to the equation rs = n(r+s) is tau(n^2), cf. A048691. Number of solutions (r,s), r>=s>0, to the equation rs = n(r+s) is (tau(n^2)+1)/2, cf. A018892.

Conjecturally, includes all positive integers except 2, 4, 7 and 11 - David W. Wilson

Examples

			There are 8 such solutions to rst = 5(r+s+t): (5, 4, 3), (7, 5, 2), (10, 4, 2), (11, 10, 1), (15, 8, 1), (20, 7, 1), (25, 3, 2), (35, 6, 1).
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• 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. A018892, A004194, A063525.

Programs

Mathematica

(* Assuming s <= 2n and t <= n*(n+2) *) redu[n_] := Reap[ Do[ red = Reduce[0 < r <= s <= t && r*s*t == n*(r+s+t), r, Integers]; If[red =!= False, Sow[{r, s, t} /. ToRules[red] ] ], {s, 1, 2*n}, {t, s, n*(n+2)}] ][[2, 1]]; a[n_] := redu[n] // Length; a[1] = 1; Table[ Print[n, " ", an = a[n]]; an, {n, 1, 75}] (* Jean-François Alcover, Feb 22 2013 *)

PARI

a(n)=sum(t=1,sqrtint(3*n),sum(s=t,sqrtint(n^2+t)+n,my(N=n*(s+t), D=s*t-n);D&&denominator(N/D)==1&&N/D>=s)) \\ Charles R Greathouse IV, Feb 22 2013

Extensions

More terms from David W. Wilson, Aug 01 2001