_Cino Hilliard (hillcino368(AT)gmail.com), _, Nov 12 2005

proposed

approved

editing

proposed

6 is the first positive squarefree composite number. 2^2 and 3^2 are the closest squares surrounding 6. So the difference, 9-4 = 5, is the first entry in the table.

closest squares surrounding 6. So the difference, 9-4 = 5, is the first entry

in the table.

approved

editing

easy,nonn,new

Cino Hilliard (hillcino368(AT)hotmailgmail.com), Nov 12 2005

Difference between the closest squares surrounding squarefree composite numbers.

5, 5, 7, 7, 7, 7, 9, 9, 9, 9, 9, 11, 11, 11, 11, 11, 11, 11, 11, 13, 13, 13, 13, 13, 13, 13, 13, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19

6,1

Let n be a squarefree composite number and r = floor(sqrt(n)). Then the closest surrounding squares of n are r^2 and (r+1)^2. So d = (r+1)^2 - r^2 = 2r+1.

6 is the first positive squarefree composite number. 2^2 and 3^2 are the

closest squares surrounding 6. So the difference, 9-4 = 5, is the first entry

in the table.

(PARI) surrsq(n) = { local(x, y, j, r, d); for(x=1, n, if(!issquare(x)&!isprime(x), r=floor(sqrt(x)); d=r+r+1; print1(d", ") \ print1(r^2", "(r+1)^2", ") ) ) }

easy,nonn

Cino Hilliard (hillcino368(AT)hotmail.com), Nov 12 2005

approved