A025285 Numbers that are the sum of 2 nonzero squares in exactly 2 ways.

50, 65, 85, 125, 130, 145, 170, 185, 200, 205, 221, 250, 260, 265, 290, 305, 338, 340, 365, 370, 377, 410, 442, 445, 450, 481, 485, 493, 500, 505, 520, 530, 533, 545, 565, 578, 580, 585, 610, 625, 629, 680, 685, 689, 697, 730, 740, 745, 754, 765, 785, 793, 800, 820

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Comments

Order and signs don't count. E.g. 50 = 5^2+5^2 = 7^2+1^2 (= (-5)^2+5^2, but that doesn't count as different).

A131574 is a subsequence. - Zak Seidov, Jan 31 2014

A025426(a(n)) = 2. - Reinhard Zumkeller, Feb 26 2015

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Square Number.
• G. Xiao, Two squares
• Index entries for sequences related to sums of squares

Crossrefs

Cf. A025284, A025286, A025287, A025288, A025289, A025290, A025291, A025292, A025293, A025426.

Cf. A007692.

Programs

Haskell

a025285 n = a025285_list !! (n-1)
a025285_list = filter ((== 2) . a025426) [1..]
-- Reinhard Zumkeller, Feb 26 2015

Mathematica

selQ[n_] := Length[ Select[ PowersRepresentations[n, 2, 2], Times @@ # != 0 &]] == 2; Select[Range[1000], selQ] (* Jean-François Alcover, Oct 03 2013 *)

PARI

is(n)=sum(k=sqrtint((n-1)\2)+1,sqrtint(n-1), issquare(n-k^2))==2 \\ Charles R Greathouse IV, May 24 2016

PARI

is(n)=my(v=valuation(n, 2), f=factor(n>>v), t=1); for(i=1, #f[, 1], if(f[i, 1]%4==1, t*=f[i, 2]+1, if(f[i, 2]%2, return(0)))); if(t%2, t-(-1)^v, t)==4 \\ Charles R Greathouse IV, May 24 2016

Formula

a(n) >= A007692(n) with equality only for n <= 16. - Alois P. Heinz, Mar 23 2023