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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.
1

%I #11 Jan 21 2024 23:43:23

%S 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 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 1,2,2,1,2,2,1,1,0,0,3,2,1,2,2,1,2,1,1,0,1,0,0

%N 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 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 Erdős and Rosenfeld (proposition 3.1) show that T(n,k) is finite.

%C 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 Diagonal T(n,n-1) = 0 since in that case M <= 0 for all d.

%C Diagonal T(n,n-2) = 0 since in that case M is not an integer for d=1 and otherwise M <= 0.

%H Kevin Ryde, <a href="/A368571/b368571.txt">Table of n, a(n) for rows n=2..150, flattened</a>

%H 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 T(n,k) = number of rows of A368312 which contain both n and k.

%e Triangle begins:

%e k=1 2 3 4 5 6 7 8

%e n=2: 0

%e n=3: 0, 0

%e n=4: 1, 0, 0

%e n=5: 1, 1, 0, 0

%e n=6: 1, 0, 1, 0, 0

%e n=7: 1, 2, 1, 1, 0, 0

%e n=8: 2, 1, 1, 0, 1, 0, 0

%e n=9: 1, 1, 2, 1, 1, 1, 0, 0

%o (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 Cf. A368312 (factor differences).

%K nonn,easy,tabl

%O 2,17

%A _Kevin Ryde_, Dec 30 2023