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A379985 Numbers k such that k^2 is of the form b^2 + (4*c)^2 where b*c is squarefree.

%I A379985 #33 Feb 03 2025 13:51:02
%S A379985 5,13,17,25,29,37,61,65,85,109,137,145,149,157,169,173,193,197,205,
%T A379985 221,229,241,265,269,293,305,325,365,377,401,409,421,433,445,485,505,
%U A379985 533,541,557,565,569,629,673,685,689,701,709,725,761,773,797
%N A379985 Numbers k such that k^2 is of the form b^2 + (4*c)^2 where b*c is squarefree.
%C A379985 It is known that the sum of squares of two odd numbers cannot be a square number, and when the sum of square of two numbers is the square of an odd number, the even one among the two numbers has to be multiple of 4. Thus the Mathematica program will not miss any entries.
%C A379985 a(n) == 1 (mod 4).
%C A379985 Numbers 4x^2 + y^2 where x, y are coprime numbers such that y is odd and x, y, 2x+y, 2x-y are squarefree. - _Yifan Xie_, Jan 09 2025, corrected by _Robert Israel_, Feb 03 2025
%H A379985 Robert Israel, <a href="/A379985/b379985.txt">Table of n, a(n) for n = 1..10000</a>
%e A379985 5 is a term since 5^2 = 3^2 + (4*1)^2 and 3*1 is squarefree.
%e A379985 149 is a term since 149^2 = 51^2 + (4*35)^2 and 51*35 = 3*5*7*17 is squarefree.
%p A379985 N:= 1000: # for terms <= N
%p A379985 Res:= {}:
%p A379985 for x from 1 while 4*x^2 < N do
%p A379985   if not numtheory:-issqrfree(x) then next fi;
%p A379985   for y from 1 by 2  while 4*x^2 + y^2 <= N do
%p A379985     if igcd(x,y) = 1 and andmap(numtheory:-issqrfree,[y,2*x+y,2*x-y]) then Res:= Res union  {4*x^2 + y^2} fi
%p A379985 od od:
%p A379985 sort(convert(Res,list)); # _Robert Israel_, Feb 03 2025
%t A379985 a = {}; Do[m = n^2; b = n; While[b = b - 2; b > 1, k = m - b^2; If[c = Sqrt[k]/4; IntegerQ[c] && SquareFreeQ[b*c], AppendTo[a, n]]], {n, 5, 800, 2}]; a
%Y A379985 Subsequence of A009003.
%Y A379985 Cf. A005117, A084645, A009000, A057100.
%K A379985 nonn
%O A379985 1,1
%A A379985 _Lei Zhou_, Jan 07 2025
%E A379985 Edited by _Robert Israel_, Feb 03 2025