OFFSET
2,17
COMMENTS
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.
Erdős and Rosenfeld (proposition 3.1) show that T(n,k) is finite.
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.
Diagonal T(n,n-1) = 0 since in that case M <= 0 for all d.
Diagonal T(n,n-2) = 0 since in that case M is not an integer for d=1 and otherwise M <= 0.
LINKS
Kevin Ryde, Table of n, a(n) for rows n=2..150, flattened
Paul Erdős and Moshe Rosenfeld, The factor-difference set of integers, Acta Arithmetica, volume 79, number 4, 1997, pages 353-359.
FORMULA
T(n,k) = number of rows of A368312 which contain both n and k.
EXAMPLE
Triangle begins:
k=1 2 3 4 5 6 7 8
n=2: 0
n=3: 0, 0
n=4: 1, 0, 0
n=5: 1, 1, 0, 0
n=6: 1, 0, 1, 0, 0
n=7: 1, 2, 1, 1, 0, 0
n=8: 2, 1, 1, 0, 1, 0, 0
n=9: 1, 1, 2, 1, 1, 1, 0, 0
PROG
(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);
CROSSREFS
KEYWORD
AUTHOR
Kevin Ryde, Dec 30 2023
STATUS
approved