A118882 Numbers which are the sum of two squares in two or more different ways.

25, 50, 65, 85, 100, 125, 130, 145, 169, 170, 185, 200, 205, 221, 225, 250, 260, 265, 289, 290, 305, 325, 338, 340, 365, 370, 377, 400, 410, 425, 442, 445, 450, 481, 485, 493, 500, 505, 520, 530, 533, 545, 565, 578, 580, 585, 610, 625, 629, 650, 676, 680

Offset

1,1

Comments

Numbers whose prime factorization includes at least two primes (not necessarily distinct) congruent to 1 mod 4 and any prime factor congruent to 3 mod 4 has even multiplicity. Products of two values in A004431.

Squares of distances that are the distance between two points in the square lattice in two or more nontrivially different ways. A quadrilateral with sides a,b,c,d has perpendicular diagonals iff a^2+c^2 = b^2+d^2. This sequence is the sums of the squares of opposite sides of such quadrilaterals, excluding kites (a=b,c=d), but including right triangles (the degenerate case d=0).

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Index entries for sequences related to sums of squares

Formula

A000161(a(n)) > 1. [Reinhard Zumkeller, Aug 16 2011]

Example

50 = 7^2 + 1^2 = 5^2 + 5^2, so 50 is in the sequence.

Mathematica

Select[Range[1000], Length[PowersRepresentations[#, 2, 2]] > 1&] (* Jean-François Alcover, Mar 02 2019 *)

Prog

(Haskell)
import Data.List (findIndices)
a118882 n = a118882_list !! (n-1)
a118882_list = findIndices (> 1) a000161_list
-- Reinhard Zumkeller, Aug 16 2011
(Python)
from itertools import count, islice
from math import prod
from sympy import factorint
def A118882_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue, 1)):
        f = factorint(n)
        if 1<int(not any(e&1 for e in f.values())) + (((m:=prod(1 if p==2 else (e+1 if p&3==1 else (e+1)&1) for p, e in f.items()))+((((~n & n-1).bit_length()&1)<<1)-1 if m&1 else 0))>>1):
            yield n
A118882_list = list(islice(A118882_gen(), 30)) # Chai Wah Wu, Sep 09 2022

Crossrefs

Cf. A004431, A009177, A085265.

Cf. A007692, A001481, A022544.

Sequence in context: A033902 A354379 A009177 * A085625 A116490 A230213

Adjacent sequences: A118879 A118880 A118881 * A118883 A118884 A118885

Keyword

nonn

Author

Franklin T. Adams-Watters, May 03 2006

Status

approved