|
%I #24 Sep 11 2022 00:40:01
%S 25,50,65,85,100,125,130,145,169,170,185,200,205,221,225,250,260,265,
%T 289,290,305,325,338,340,365,370,377,400,410,425,442,445,450,481,485,
%U 493,500,505,520,530,533,545,565,578,580,585,610,625,629,650,676,680
%N Numbers which are the sum of two squares in two or more different ways.
%C 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.
%C 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).
%H Reinhard Zumkeller, <a href="/A118882/b118882.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A000161(a(n)) > 1. [_Reinhard Zumkeller_, Aug 16 2011]
%e 50 = 7^2 + 1^2 = 5^2 + 5^2, so 50 is in the sequence.
%t Select[Range[1000], Length[PowersRepresentations[#, 2, 2]] > 1&] (* _Jean-François Alcover_, Mar 02 2019 *)
%o (Haskell)
%o import Data.List (findIndices)
%o a118882 n = a118882_list !! (n-1)
%o a118882_list = findIndices (> 1) a000161_list
%o -- _Reinhard Zumkeller_, Aug 16 2011
%o (Python)
%o from itertools import count, islice
%o from math import prod
%o from sympy import factorint
%o def A118882_gen(startvalue=1): # generator of terms >= startvalue
%o for n in count(max(startvalue,1)):
%o f = factorint(n)
%o 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):
%o yield n
%o A118882_list = list(islice(A118882_gen(),30)) # _Chai Wah Wu_, Sep 09 2022
%Y Cf. A004431, A009177, A085265.
%Y Cf. A007692, A001481, A022544.
%K nonn
%O 1,1
%A _Franklin T. Adams-Watters_, May 03 2006
|
