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.

4, 12, 24, 36, 40, 60, 72, 84, 112, 144, 144, 180, 180, 220, 252, 264, 312, 360, 364, 432, 420, 504, 480, 540, 544, 612, 684, 792, 760, 864, 900, 840, 936, 924, 1080, 1012, 1104, 1260, 1260, 1200, 1300, 1440, 1404, 1584, 1512, 1764, 1624, 1836, 1740, 1860

Offset

1,1

Comments

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

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

John F. Richardson, A Property of Odd Composites, Math Forums, 2012.

Example

15=7+8 (partition is x,x+1)
col 1 sum(to products)
1*6=6
2*5=10
3*4=12
col 2 sum(to products)
1*7=7
2*6=12
3*5=15
4*4=16
There is an overlapping product, and the lowest is 12.
This indicates the original N of 15 is composite.

Prog

(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]
forstep(n=9, 300, 2, if(!isprime(n), print1(do(n)", "))) \\ Charles R Greathouse IV, Oct 28 2012

Crossrefs

Cf. A071904.

Sequence in context: A301265 A285350 A072389 * A297568 A353795 A317518

Adjacent sequences: A218388 A218389 A218390 * A218392 A218393 A218394

Keyword

nonn

Author

Bill McEachen, Oct 27 2012

Status

approved