A368312 - OEIS

%I #13 Jan 21 2024 23:35:39

%S 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,

%T 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,

%U 13,29,30,4,14,31,8,32,15,33,2,34,0,5,9,16,35,36,17,37

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

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

%C 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.

%C Row n has length A038548(n).

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

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

%H Kevin Ryde, <a href="/A368312/b368312.txt">Table of n, a(n) for rows n=1..2500, flattened</a>

%H Paul Erdős and Moshe Rosenfeld, <a href="https://doi.org/10.4064/aa-79-4-353-359">The factor-difference set of integers</a>, Acta Arithmetica, volume 79, number 4, 1997, pages 353-359.

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

%e Triangle begins:

%e        k=1

%e   n=1:   0

%e   n=2:   1

%e   n=3:   2

%e   n=4:   0, 3

%e   n=5:   4

%e   n=6:   1, 5

%e   n=7:   6

%e   n=8:   2, 7

%e   n=9:   0, 8

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

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

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

%Y Cf. A335572 (factor sums).

%K nonn,easy,tabf

%O 1,3

%A _Kevin Ryde_, Dec 21 2023