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.

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, 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, 1, 2, 2, 1, 2, 2, 1, 1, 0, 0, 3, 2, 1, 2, 2, 1, 2, 1, 1, 0, 1, 0, 0

Offset: 2

Author

Kevin Ryde, Dec 30 2023

Keywords

• nonn
• easy
• tabl

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.

Examples

			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
		

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.

Crossrefs

Cf. A368312 (factor differences).

Programs

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);

Formula

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