A180416 Number of positive integers below 10^n, excluding perfect squares, which have a representation as a sum of 2 positive squares.

3, 33, 298, 2649, 23711, 215341, 1982296, 18447847, 173197435, 1637524156, 15570196516, 148735628858, 1426303768587, 13722207893214, 132387231596281, 1280309591127436

Offset: 1

Author

Martin Renner, Jan 19 2011

Keywords

• nonn
• more

Comments

Numbers that can be represented as a sum of three or more positive squares but not as a sum of two positive squares (e.g., 3=1^2+1^2+1^2 or 6=1^2+1^2+2^2) are not counted. Numbers that can be represented as a sum of two positive squares and alternatively as a sum of three or more positive squares are counted (e.g., 18 = 9+9 = 1+1+16, 26, 41, ...).

Links

• Eric W. Weisstein: MathWorld -- Lagrange's Four-Square Theorem.
• Eric W. Weisstein: MathWorld -- Sum of Squares Function.

Crossrefs

Cf. A000415, A000404, A000290, A002283, A049416, A167613, A180425.

Programs

Maple

isA000415 := proc(n) local x ,y2; if issqr(n) then false; else for x from 1 do y2 := n-x^2 ; if y2 < x^2 then return false; elif issqr(y2) then return true; end if; end do ; end if; end proc:
A180416 := proc(n) a := 0 ; for k from 2 to 10^n-1 do if isA000415(k) then a := a+1 ; end if; end do: a ; end proc:
for n from 1 do print(A180416(n)) ; end do; # R. J. Mathar, Jan 20 2011

Mathematica

a[n_] := a[n] = Module[{k, xMax = Floor[Sqrt[10^n - 1]]}, Table[k = x^2 + y^2; If[IntegerQ[Sqrt[k]], Nothing, k], {x, 1, xMax}, {y, x, Floor[ Sqrt[10^n - 1 - x^2]]}] // Flatten // Union // Length];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 8}] (* Jean-François Alcover, Oct 31 2020 *)

Formula

a(n) = |{ 0A000415} }|.

a(n) = |{ 0A000404} \ {A000290}) }|.

a(n) = A002283(n) - A049416(n) - A167615(n) - A180425(n).

Extensions

a(6)-a(8) from Alois P. Heinz, Jan 20 2011

a(9)-a(10) from Donovan Johnson, Feb 04 2011

a(10) corrected and a(11)-a(16) from Hiroaki Yamanouchi, Jul 13 2014