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A218391 Let k be the n-th odd composite, then a(n) is the smallest w*x such that w + x = (k-1)/2, y + z = (k+1)/2, and w*x = y*z.

%I A218391 #36 Feb 17 2025 03:30:33
%S A218391 4,12,24,36,40,60,72,84,112,144,144,180,180,220,252,264,312,360,364,
%T A218391 432,420,504,480,540,544,612,684,792,760,864,900,840,936,924,1080,
%U A218391 1012,1104,1260,1260,1200,1300,1440,1404,1584,1512,1764,1624,1836,1740,1860
%N A218391 Let k be the n-th odd composite, then a(n) is the smallest w*x such that w + x = (k-1)/2, y + z = (k+1)/2, and w*x = y*z.
%C A218391 If a number w + x + y + z with w, x, y, z > 0 has w*x = y*z then it is composite.
%C A218391 Appears to be a subset of A072389. - _Bill McEachen_, Feb 14 2025
%H A218391 Charles R Greathouse IV, <a href="/A218391/b218391.txt">Table of n, a(n) for n = 1..10000</a>
%H A218391 John F. Richardson, <a href="https://mathforums.com/threads/a-property-of-odd-composites.27194/">A Property of Odd Composites</a>, Math Forums, 2012.
%e A218391 15=7+8 (partition is x,x+1)
%e A218391 col 1 sum(to products)
%e A218391 1*6=6
%e A218391 2*5=10
%e A218391 3*4=12
%e A218391 col 2 sum(to products)
%e A218391 1*7=7
%e A218391 2*6=12
%e A218391 3*5=15
%e A218391 4*4=16
%e A218391 There is an overlapping product, and the lowest is 12.
%e A218391 This indicates the original N of 15 is composite.
%o A218391 (PARI) do(n)=my(X=vector(n\4,i,i*(n\2-i)),Y=vector((n+1)\4,i,i*(n\2-i+1)),i=1,j=1);while(X[i]!=Y[j],if(X[i]<Y[j],i++,j++));X[i]
%o A218391 forstep(n=9,300,2,if(!isprime(n),print1(do(n)", "))) \\ _Charles R Greathouse IV_, Oct 28 2012
%Y A218391 Cf. A071904.
%K A218391 nonn
%O A218391 1,1
%A A218391 _Bill McEachen_, Oct 27 2012