cp's OEIS Frontend

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

%I A063123 #14 Aug 11 2025 10:57:07
%S A063123 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,
%T A063123 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,
%U A063123 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
%N A063123 Number of solutions (r,s), 0< r< s, to the equation 1/n = 1/r + 1/s + 1/(r*s).
%C A063123 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.
%C A063123 Number of divisors of n-th oblong number not greater than n. - Chandler
%H A063123 Ray Chandler, <a href="/A063123/b063123.txt">Table of n, a(n) for n=1..10000</a>
%F A063123 a(n) = tau(n)*tau(n+1)/2 = A092517(n)/2.
%e A063123 a(2)=2 because 1/2=1/3+1/8+1/24=1/4+1/5+1/20.
%t A063123 a[n_]:=DivisorSigma[0,n]DivisorSigma[0,(n+1)]/2; Array[a,86] (* _Stefano Spezia_, Aug 11 2025 *)
%o A063123 (PARI) a(n) = numdiv(n)*numdiv(n+1)/2 \\ _Michel Marcus_, Jun 17 2013
%Y A063123 Cf. A063520.
%K A063123 nonn
%O A063123 1,2
%A A063123 _Vladeta Jovovic_, Aug 08 2001