cp's OEIS Frontend

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.

%I A172412 #10 Mar 30 2012 18:52:04
%S A172412 0,4,8,12,16,24,32,36,40,48,60,64,72,80,84,96,100,112,120,128,140,144,
%T A172412 160,168,180,192,196,200,216,224,240,252,256,264,280,288,308,320,324,
%U A172412 336,352,360,364,384,396,400,416,432,440,448,468,480,484
%N 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.
%C A172412 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.
%C A172412 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,...
%C A172412 The locations of the -1 are in A016825.
%C A172412 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.
%C A172412 The sequence shows the arguments 4n of this quadrisection for which S(4n) is not larger than any S(4n') for n'>n.
%C A172412 Evidently, the sequence is a subsequence of A008586.
%e A172412 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.
%p A172412 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:
%p A172412 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:
%p A172412 for n from 0 to 500 by 4 do if isA172412(n) then printf("%d,",n); end if; end do:
%K A172412 nonn
%O A172412 1,2
%A A172412 _Paul Curtz_, Nov 20 2010