A368312 Irregular triangle read by rows where row n lists the factor differences of n.

0, 1, 2, 0, 3, 4, 1, 5, 6, 2, 7, 0, 8, 3, 9, 10, 1, 4, 11, 12, 5, 13, 2, 14, 0, 6, 15, 16, 3, 7, 17, 18, 1, 8, 19, 4, 20, 9, 21, 22, 2, 5, 10, 23, 0, 24, 11, 25, 6, 26, 3, 12, 27, 28, 1, 7, 13, 29, 30, 4, 14, 31, 8, 32, 15, 33, 2, 34, 0, 5, 9, 16, 35, 36, 17, 37

Offset: 1

Author

Kevin Ryde, Dec 21 2023

Keywords

• nonn
• easy
• tabf

Comments

Factor differences of n are all abs(p-q) where n = p*q, for positive integers p,q.

p is each divisor of n which is >= sqrt(n), in ascending order (A161908), and the resulting differences p-q are distinct and in ascending order.

Row n has length A038548(n).

Row n begins with smallest difference T(n,1) = A056737(n) and this is 0 iff n is a perfect square.

Row n ends with n-1 and this is the sole entry iff n is 1 or prime.

Examples

			Triangle begins:
       k=1
  n=1:   0
  n=2:   1
  n=3:   2
  n=4:   0, 3
  n=5:   4
  n=6:   1, 5
  n=7:   6
  n=8:   2, 7
  n=9:   0, 8
		

Links

• Kevin Ryde, Table of n, a(n) for rows n=1..2500, flattened
• Paul Erdős and Moshe Rosenfeld, The factor-difference set of integers, Acta Arithmetica, volume 79, number 4, 1997, pages 353-359.

Crossrefs

Cf. A038548 (row lengths), A079667 (row sums), A068333 (row products).

Cf. A056737 (column k=1), A161908.

Cf. A335572 (factor sums).

Programs

PARI

row(n) = my(v=divisors(n)); (v-Vecrev(v))[#v\2+1..#v];

Formula

T(n,k) = d - n/d where d = A161908(n,k).