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

2,17

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

Cf. A368312 (factor differences).

Sequence in context: A353129 A117452 A029412 * A178670 A026921 A198568

Adjacent sequences: A368568 A368569 A368570 * A368572 A368573 A368574

Keyword

nonn,easy,tabl

Author

Kevin Ryde, Dec 30 2023

Status

approved