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A081324 Twice a square but not the sum of 2 distinct squares.

%I A081324 #104 Sep 15 2019 17:47:28
%S A081324 0,2,8,18,32,72,98,128,162,242,288,392,512,648,722,882,968,1058,1152,
%T A081324 1458,1568,1922,2048,2178,2592,2888,3528,3698,3872,4232,4418,4608,
%U A081324 4802,5832,6272,6498,6962,7688,7938,8192,8712,8978,9522,10082,10368,11552
%N A081324 Twice a square but not the sum of 2 distinct squares.
%C A081324 Conjecture: for n>1 this is A050804.
%C A081324 From _Altug Alkan_, Apr 12 2016: (Start)
%C A081324 Conjecture is true. Proof :
%C A081324 If n = a^2 + b^2, where a and b are nonzero integers, then n^3 = (a^2 + b^2)^3 = A^2 + B^2 = C^2 + D^2 where;
%C A081324 A = 2*a^2*b + (a^2-b^2)*b = 3*a^2*b - b^3,
%C A081324 B = 2*a*b^2 - (a^2-b^2)*a = 3*a*b^2 - a^3,
%C A081324 C = 2*a*b^2 + (a^2-b^2)*a = 1*a*b^2 + a^3,
%C A081324 D = 2*a^2*b - (a^2-b^2)*b = 1*a^2*b + b^3.
%C A081324 Obviously, A, B, C, D are always nonzero because a and b are nonzero integers. Additionally, if a^2 is not equal to b^2, then (A, B) and (C, D) are distinct pairs, that is, n^3 can be expressible as a sum of two nonzero squares more than one way. Since we know that n is a sum of two nonzero squares if and only if n^3 is a sum of two nonzero squares (see comment section of A000404); if n^3 is the sum of two nonzero squares in exactly one way, n must be a^2 + b^2 with a^2 = b^2 and n is the sum of two nonzero squares in exactly one way. That is the definition of this sequence, so this sequence is exactly A050804 except "0" that is the first term of this sequence. (End) [Edited by _Altug Alkan_, May 14 2016]
%C A081324 Conjecture: sequence consists of numbers of form 2*k^2 such that sigma(2*k^2)==3 (mod 4) and k is not divisible by 5.
%C A081324 The reason of related observation is that 5 is the least prime of the form 4*m+1. However, counterexamples can be produced. For example 57122 = 2*169^2 and sigma(57122) == 3 (mod 4) and it is not divisible by 5. - _Altug Alkan_, Jun 10 2016
%C A081324 For n > 0, this sequence lists numbers n such that n is the sum of two nonzero squares while n^2 is not. - _Altug Alkan_, Apr 11 2016
%C A081324 2*k^2 where k has no prime factor == 1 (mod 4). - _Robert Israel_, Jun 10 2016
%H A081324 Evan M. Bailey, <a href="/A081324/b081324.txt">Table of n, a(n) for n = 1..20000</a> (first 115 terms from Reinhard Zumkeller, terms 116-1000 from Zak Seidov)
%H A081324 Evan M. Bailey, <a href="/A081324/a081324.cpp.txt">a081324.cpp</a>
%F A081324 A063725(a(n)) = 1. [_Reinhard Zumkeller_, Aug 17 2011]
%F A081324 a(n) = 2*A004144(n-1)^2 for n > 1. - _Charles R Greathouse IV_, Jun 18 2013
%p A081324 map(k -> 2*k^2, select(k -> andmap(t -> t[1] mod 4 <> 1, ifactors(k)[2]), [$0..100])); # _Robert Israel_, Jun 10 2016
%t A081324 Select[ Range[0, 12000], MatchQ[ PowersRepresentations[#, 2, 2], {{n_, n_}}] &] (* _Jean-François Alcover_, Jun 18 2013 *)
%o A081324 (Haskell)
%o A081324 import Data.List (elemIndices)
%o A081324 a081324 n = a081324_list !! (n-1)
%o A081324 a081324_list = 0 : elemIndices 1 a063725_list
%o A081324 -- _Reinhard Zumkeller_, Aug 17 2011
%o A081324 (PARI) concat([0,2],apply(n->2*n^2, select(n->vecmin(factor(n)[, 1]%4)>1, vector(100,n,n+1)))) \\ _Charles R Greathouse IV_, Jun 18 2013
%Y A081324 Cf. A050804, A025284, A063725, A125022, A018825, A000404, A004431.
%K A081324 nonn
%O A081324 1,2
%A A081324 _Benoit Cloitre_, Apr 20 2003
%E A081324 a(19)-a(45) from _Donovan Johnson_, Nov 15 2009
%E A081324 Offset corrected by _Reinhard Zumkeller_, Aug 17 2011