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A368571 Triangle read by rows where T(n,k) is the number of positive integers M which have both n and k as factor differences, 1 <= k < n.

%I A368571 #11 Jan 21 2024 23:43:23
%S A368571 0,0,0,1,0,0,1,1,0,0,1,0,1,0,0,1,2,1,1,0,0,2,1,1,0,1,0,0,1,1,2,1,1,1,
%T A368571 0,0,2,1,1,1,2,0,1,0,0,3,2,1,3,1,1,1,1,0,0,1,1,3,1,1,1,1,0,1,0,0,3,3,
%U A368571 1,2,2,1,2,2,1,1,0,0,3,2,1,2,2,1,2,1,1,0,1,0,0
%N A368571 Triangle read by rows where T(n,k) is the number of positive integers M which have both n and k as factor differences, 1 <= k < n.
%C A368571 The factor differences of some M are all abs(p-q) where M = p*q for positive integers p,q, being row M of A368312.
%C A368571 Erdős and Rosenfeld (proposition 3.1) show that T(n,k) is finite.
%C A368571 Their method shows the relevant M are those M = (d^2 + (G/d)^2 - 2*(n^2+k^2))/16 which are positive integers, for G = n^2 - k^2, d < sqrt(G), and d divides G.
%C A368571 Diagonal T(n,n-1) = 0 since in that case M <= 0 for all d.
%C A368571 Diagonal T(n,n-2) = 0 since in that case M is not an integer for d=1 and otherwise M <= 0.
%H A368571 Kevin Ryde, <a href="/A368571/b368571.txt">Table of n, a(n) for rows n=2..150, flattened</a>
%H A368571 Paul Erdős and Moshe Rosenfeld, <a href="https://doi.org/10.4064/aa-79-4-353-359">The factor-difference set of integers</a>, Acta Arithmetica, volume 79, number 4, 1997, pages 353-359.
%F A368571 T(n,k) = number of rows of A368312 which contain both n and k.
%e A368571 Triangle begins:
%e A368571       k=1  2  3  4  5  6  7  8
%e A368571   n=2:  0
%e A368571   n=3:  0, 0
%e A368571   n=4:  1, 0, 0
%e A368571   n=5:  1, 1, 0, 0
%e A368571   n=6:  1, 0, 1, 0, 0
%e A368571   n=7:  1, 2, 1, 1, 0, 0
%e A368571   n=8:  2, 1, 1, 0, 1, 0, 0
%e A368571   n=9:  1, 1, 2, 1, 1, 1, 0, 0
%o A368571 (PARI) T(n,k) = my(t=2*(n^2+k^2), v=apply(sqr,divisors(n^2-k^2))); sum(i=1,#v\2, my(m=v[i]+v[#v-i+1]-t); m>0 && m%16==0);
%Y A368571 Cf. A368312 (factor differences).
%K A368571 nonn,easy,tabl
%O A368571 2,17
%A A368571 _Kevin Ryde_, Dec 30 2023