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

1,2

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.

Links

Table of n, a(n) for n=1..53.

Example

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.

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:

Crossrefs

Sequence in context: A311392 A311393 A311394 * A322136 A212502 A071385

Adjacent sequences: A172409 A172410 A172411 * A172413 A172414 A172415

Keyword

nonn

Author

Paul Curtz, Nov 20 2010

Status

approved