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A278648 Consider the set S of integers 1 through n. a(n) is the number of unordered ways in which three distinct elements {a, b, c} of S satisfy a*b = c*n.

%I A278648 #23 May 22 2025 04:22:56
%S A278648 0,0,0,0,0,0,2,0,2,1,4,0,8,0,6,8,7,0,13,0,16,12,10,0,26,6,12,13,24,0,
%T A278648 38,0,23,20,16,24,46,0,18,24,50,0,56,0,40,49,22,0,71,15,46,32,48,0,67,
%U A278648 40,74,36,28,0,120,0,30,73,61,48,92,0,64,44,106,0,136,0,36,86,72,60,110
%N A278648 Consider the set S of integers 1 through n. a(n) is the number of unordered ways in which three distinct elements {a, b, c} of S satisfy a*b = c*n.
%C A278648 Inspired by A278348.
%C A278648 Index of first occurrence of k >= 0, or zero if no such number exists: 1, 9, 6, 0, 10, 0, 14, 16, 12, 0, 22, 0, 21, 18, 0, 49, 20, 0, 38, 0, 33, 0, 46, 32, 28, 0, ..., ;
%C A278648 Numbers which never occur: 3, 5, 9, 11, 14, 17, 19, 21, 25, 27, 29, 31, 33, 34, 35, 37, 39, 41, 43, 47, 51, ..., ;
%C A278648 Records: 0, 2, 4, 8, 13, 16, 26, 38, 46, 50, 56, 71, 74, 120, 136, 176, 193, 214, 330, 355, 482, 574, 668, 839, 890, 996, 1088, 1223, 1528, 1920, 2039, 2224, 2374, 2646, 3055, 3120, 3811, 5010, 5539, 6208, 6591, 8566, 9139, 9690, 12359, 13894, 14796, 15331, 16118, 16558, 22048, ..., ;
%C A278648 which first occur for n: 0, 6, 10, 12, 18, 20, 24, 30, 36, 40, 42, 48, 56, 60, 72, 84, 90, 108, 120, 144, 168, 180, 210, 240, 280, 300, 330, 336, 360, 420, 480, 504, 540, 600, 630, 660, 720, 840, 1008, 1080, 1200, 1260, 1440, 1560, 1680, 1980, 2100, 2160, 2310, 2340, 2520, ..., .
%C A278648 If instead we look for the number of unordered ways two distinct elements {a, b} of S satisfy a*b = n, then a(n) = floor(sigma_0(n) - 2)) = A211159(n+1).
%C A278648 Number of 2 X 2 singular matrices of the form
%C A278648   [c a]
%C A278648   [b n]
%C A278648 with a, b, and c distinct positive integers less than n and a < b.
%H A278648 Bobby Jacobs and Robert G. Wilson v, <a href="/A278648/b278648.txt">Table of n, a(n) for n = 0..10000</a>
%F A278648 a(n) = (A278348(n) - A278348(n-1))/8.
%F A278648 a(p) = 0 for any prime p and for n: 0, 1 & 4.
%e A278648 a(6) = 2 since 2*3 = 1*6 and 3*4 = 2*6;
%e A278648 a(8) = 2 since 2*4 = 1*8 and 4*6 = 3*8;
%e A278648 a(9) = 1 since 3*6 = 2*9;
%e A278648 a(10) = 4 since 2*5 = 1*10, 4*5 = 2*10, 5*6 = 3*10, and 5*8 = 4*10;
%e A278648 a(12) = 8 since 2*6 = 1*12, 3*4 = 1*12, 3*8 = 2*12, 4*6 = 2*12, 4*9 = 3*12, 6*8 = 4*12, 6*10 = 5*12, and 8*9 = 6*12;
%e A278648 etc.
%t A278648 f[n_] := Block[{c = 0, k = 1}, While[k < n, c += Count[ Times @@@ Select[ Tuples[ Rest@ Most@ Divisors[k*n], 2], #[[1]] < #[[2]] < n &], k*n]; k++]; c]; Array[f, 52]
%Y A278648 Cf. A278348.
%K A278648 nonn,easy
%O A278648 0,7
%A A278648 _Bobby Jacobs_ and _Robert G. Wilson v_, Nov 25 2016