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

1,3

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.

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.

Formula

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

Example

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

Prog

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

Crossrefs

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

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

Cf. A335572 (factor sums).

Sequence in context: A352579 A241319 A287016 * A285722 A274441 A213859

Adjacent sequences: A368309 A368310 A368311 * A368313 A368314 A368315

Keyword

nonn,easy,tabf

Author

Kevin Ryde, Dec 21 2023

Status

approved