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A368312
Irregular triangle read by rows where row n lists the factor differences of n.
3
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
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
KEYWORD
nonn,easy,tabf
AUTHOR
Kevin Ryde, Dec 21 2023
STATUS
approved