A172412 Multiples of 4 with the property that addition of a square gives a square that is not larger than the square for any later term.

0, 4, 8, 12, 16, 24, 32, 36, 40, 48, 60, 64, 72, 80, 84, 96, 100, 112, 120, 128, 140, 144, 160, 168, 180, 192, 196, 200, 216, 224, 240, 252, 256, 264, 280, 288, 308, 320, 324, 336, 352, 360, 364, 384, 396, 400, 416, 432, 440, 448, 468, 480, 484

Offset: 1

Author

Paul Curtz, Nov 20 2010

Keywords

• nonn

Comments

For each integer n one can define smallest-next-square function S(n) such that S(n) is a perfect square, S(n) >=n, and such that S(n)-n is also a perfect square. If there are more such S(n), take the smallest, use -1 if this S(n) does not exist.

This S(n) starts for n>=0 as 0, 1, -1, 4, 4, 9, -1, 16, 9, 9, -1, 36, 16, 49, -1, 16, 16, 81,...

The locations of the -1 are in A016825.

The quadrisection S(4n) is S(0)=0, S(4)=4, S(8)=9, S(12)=16, S(16)=16, S(20)=36, S(24)=25, S(28)=64.

The sequence shows the arguments 4n of this quadrisection for which S(4n) is not larger than any S(4n') for n'>n.

Evidently, the sequence is a subsequence of A008586.

Examples

			S(20) =36 is larger than S(24), so 20 is not in the sequence. S(28)=64 is larger than S(32)=36, so 28 is not in the sequence.
		

Programs

Maple

nxtSqr := proc(n) local d,y,x ; for d in sort(convert(numtheory[divisors](n),list)) do if d >= n/d then y := (d+n/d)/2 ; if type(y,'integer') then x := d-y ; if n+x^2 = y^2 then return y^2 ; end if; end if; end if; end do: return -1 ; end proc:
isA172412 := proc(n) local y2; if n mod 4 = 0 then y2 := nxtSqr(n) ; for a from n+4 by 4 do if a > y2 then return true; elif nxtSqr(a) < y2 then return false; end if; end do: else false; end if; end proc:
for n from 0 to 500 by 4 do if isA172412(n) then printf("%d,",n); end if; end do: