A111205 Squarefree numbers n such that the difference between the closest squares surrounding n and n have a common divisor greater than 1.

14, 18, 21, 24, 33, 39, 50, 51, 54, 55, 57, 60, 63, 68, 95, 102, 105, 108, 111, 112, 114, 117, 119, 120, 138, 145, 150, 155, 160, 165, 171, 174, 177, 180, 183, 186, 189, 192, 195, 203, 248, 258, 261, 264, 267, 270, 273, 275, 276, 279, 282, 285, 286, 288, 290

Offset

6,1

Comments

Conjecture: The number of terms in this sequence is infinite.

Links

Table of n, a(n) for n=6..60.

Formula

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. If gcd(n, d) > 1 then list n.

Example

14 is a squarefree composite number. 3^2 and 4^2 are the closest squares surrounding 14. So the difference, 16-9 = 7 and 14 have a common divisor greater than 1 namely 7, so 14 is the first entry in the table.

Prog

(PARI) surrsqgcd(n) = { local(x, y, j, r, d); for(x=1, n, if(!issquare(x)&!isprime(x), r=floor(sqrt(x)); d=r+r+1; if(gcd(x, d) > 1, print1(x", ") ) )) }

Crossrefs

Sequence in context: A052026 A317743 A118499 * A097324 A051419 A000053

Adjacent sequences: A111202 A111203 A111204 * A111206 A111207 A111208

Keyword

easy,nonn

Author

Cino Hilliard, Nov 12 2005

Status

approved