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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. 1

%I #26 May 23 2020 02:11:33

%S 4,12,24,36,40,60,72,84,112,144,144,180,180,220,252,264,312,360,364,

%T 432,420,504,480,540,544,612,684,792,760,864,900,840,936,924,1080,

%U 1012,1104,1260,1260,1200,1300,1440,1404,1584,1512,1764,1624,1836,1740,1860

%N 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 If a number w + x + y + z with w, x, y, z > 0 has w*x = y*z then it is composite.

%H Charles R Greathouse IV, <a href="/A218391/b218391.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 15=7+8 (partition is x,x+1)

%e col 1 sum(to products)

%e 1*6=6

%e 2*5=10

%e 3*4=12

%e col 2 sum(to products)

%e 1*7=7

%e 2*6=12

%e 3*5=15

%e 4*4=16

%e There is an overlapping product, and the lowest is 12.

%e This indicates the original N of 15 is composite.

%o (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 forstep(n=9,300,2,if(!isprime(n),print1(do(n)", "))) \\ _Charles R Greathouse IV_, Oct 28 2012

%Y Cf. A071904.

%K nonn

%O 1,1

%A _Bill McEachen_, Oct 27 2012

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