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

%I A368312 #13 Jan 21 2024 23:35:39
%S A368312 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 A368312 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 A368312 13,29,30,4,14,31,8,32,15,33,2,34,0,5,9,16,35,36,17,37
%N A368312 Irregular triangle read by rows where row n lists the factor differences of n.
%C A368312 Factor differences of n are all abs(p-q) where n = p*q, for positive integers p,q.
%C A368312 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 A368312 Row n has length A038548(n).
%C A368312 Row n begins with smallest difference T(n,1) = A056737(n) and this is 0 iff n is a perfect square.
%C A368312 Row n ends with n-1 and this is the sole entry iff n is 1 or prime.
%H A368312 Kevin Ryde, <a href="/A368312/b368312.txt">Table of n, a(n) for rows n=1..2500, flattened</a>
%H A368312 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 A368312 T(n,k) = d - n/d where d = A161908(n,k).
%e A368312 Triangle begins:
%e A368312        k=1
%e A368312   n=1:   0
%e A368312   n=2:   1
%e A368312   n=3:   2
%e A368312   n=4:   0, 3
%e A368312   n=5:   4
%e A368312   n=6:   1, 5
%e A368312   n=7:   6
%e A368312   n=8:   2, 7
%e A368312   n=9:   0, 8
%o A368312 (PARI) row(n) = my(v=divisors(n)); (v-Vecrev(v))[#v\2+1..#v];
%Y A368312 Cf. A038548 (row lengths), A079667 (row sums), A068333 (row products).
%Y A368312 Cf. A056737 (column k=1), A161908.
%Y A368312 Cf. A335572 (factor sums).
%K A368312 nonn,easy,tabf
%O A368312 1,3
%A A368312 _Kevin Ryde_, Dec 21 2023