A063123 Number of solutions (r,s), 0< r< s, to the equation 1/n = 1/r + 1/s + 1/(r*s).

1, 2, 3, 3, 4, 4, 4, 6, 6, 4, 6, 6, 4, 8, 10, 5, 6, 6, 6, 12, 8, 4, 8, 12, 6, 8, 12, 6, 8, 8, 6, 12, 8, 8, 18, 9, 4, 8, 16, 8, 8, 8, 6, 18, 12, 4, 10, 15, 9, 12, 12, 6, 8, 16, 16, 16, 8, 4, 12, 12, 4, 12, 21, 14, 16, 8, 6, 12, 16, 8, 12, 12, 4, 12, 18, 12, 16, 8, 10, 25, 10, 4, 12, 24, 8, 8

Offset: 1

Author

Vladeta Jovovic, Aug 08 2001

Keywords

• nonn

Comments

Unordered solutions to the equation 1/n = 1/r+1/s+1/(r*s) are r=d+n, s=n*(n+1)/d+n, where d is factor of n*(n+1) not greater than n.

Number of divisors of n-th oblong number not greater than n. - Chandler

Examples

			a(2)=2 because 1/2=1/3+1/8+1/24=1/4+1/5+1/20.
		

Links

• Ray Chandler, Table of n, a(n) for n=1..10000

Crossrefs

Cf. A063520.

Programs

Mathematica

a[n_]:=DivisorSigma[0,n]DivisorSigma[0,(n+1)]/2; Array[a,86] (* Stefano Spezia, Aug 11 2025 *)

PARI

a(n) = numdiv(n)*numdiv(n+1)/2 \\ Michel Marcus, Jun 17 2013

Formula

a(n) = tau(n)*tau(n+1)/2 = A092517(n)/2.