cp's OEIS Frontend

A153193 a(n) is the number of integers of the form n*(n+1)*k / (k - n*(n+1)) where k is an integer >= 1.

%I A153193 #18 Oct 21 2024 17:28:32
%S A153193 4,13,22,22,40,40,31,52,67,40,67,67,40,121,121,40,67,67,67,202,121,40,
%T A153193 94,157,67,94,157,67,121,121,49,148,121,121,337,112,40,121,283,94,121,
%U A153193 121,67,337,202,40,121,202,112,202,202,67,94,283,283,283,121,40
%N A153193 a(n) is the number of integers of the form n*(n+1)*k / (k - n*(n+1)) where k is an integer >= 1.
%C A153193 1/n - 1/(n+1) - 1/k = 1/c where c is an integer, k >= 1.
%H A153193 Robert Israel, <a href="/A153193/b153193.txt">Table of n, a(n) for n = 1..10000</a>
%e A153193 The a(1)=4 integers of the form n*(n+1)*k/(k - n*(n+1)) = 1*(1+1)*k/(k - 1*(1+1)) = 2*k/(k-2) occur at
%e A153193   k=1: 2*1/(1-2) = -2,
%e A153193   k=3: 2*3/(3-2) =  6,
%e A153193   k=4: 2*4/(4-2) =  4, and
%e A153193   k=6: 2*6/(6-2) =  3.
%p A153193 f:= proc(n) local D;
%p A153193    D:= numtheory:-divisors((n*(n+1))^2);
%p A153193    nops(D) + nops(select(`<=`,D,n*(n+1)-1))
%p A153193 end proc:
%p A153193 map(f, [$1..100]); # _Robert Israel_, Oct 21 2024
%Y A153193 Cf. A146564, A152492, A063647, A000005, A146566.
%K A153193 nonn
%O A153193 1,1
%A A153193 _Ctibor O. Zizka_, Dec 20 2008
%E A153193 a(13)-a(58) from _Jon E. Schoenfield_, Mar 15 2022