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%I #74 Mar 21 2023 07:34:02
%S 1,2,3,3,3,6,3,4,5,6,3,9,3,6,9,5,3,10,3,9,9,6,3,12,5,6,7,9,3,18,3,6,9,
%T 6,9,15,3,6,9,12,3,18,3,9,15,6,3,15,5,10,9,9,3,14,9,12,9,6,3,27,3,6,
%U 15,7,9,18,3,9,9,18,3,20,3,6,15,9,9,18,3,15,9,6,3,27,9,6,9,12,3,30,9,9,9,6,9
%N a(n) = tau(2*n^2)/2.
%C Inverse Moebius transform of A068068. Number of elements in the set {(x,y): x is odd, x|n, y|n, gcd(x,y)=1}.
%C The number of Pythagorean points (x,y), 0 < x < y, located on the hyperbola y = 2n(x-n)/(x-2n) and having "excess" x+y-z = 2n. - _Seppo Mustonen_, Jun 07 2005
%C a(n) is the number of Pythagorean triangles with radius of the inscribed circle equal to n. For number of primitive Pythagorean triangles having inradius n, see A068068(n). - _Ant King_, Mar 06 2006
%C Dirichlet convolution of A048691 and A154269. - _R. J. Mathar_, Jun 01 2011
%C Number of distinct L-shapes of thickness n where the L area equals the rectangular area that it "contains". Visually can be thought as those areas of A156688 (surrounded by equal border of thickness n: 2xy = (x+2n)(y+2n), x and y positive integers) where both x and y are even, so they can be split into L-shapes. So L-shapes have formula: 2xy = (x+n)(y+n). - _Juhani Heino_, Jul 23 2012
%H Vincenzo Librandi, <a href="/A078644/b078644.txt">Table of n, a(n) for n = 1..10000</a>
%H S. Mustonen, <a href="http://www.survo.fi/papers/pythagorean.pdf">Visualization and characterization of Pythagorean triples</a>
%H Seppo Mustonen, <a href="/A078644/a078644.pdf">Visualization and characterization of Pythagorean triples</a> [Local copy]
%H T. Omland, <a href="http://dx.doi.org/10.1016/j.jnt.2016.06.009">How many Pythagorean triples with given inradius?</a>, J. Numb. Theory 170 (2017) 1-2.
%F Multiplicative with a(2^e) = e+1, a(p^e) = 2*e+1, p > 2. a(n) = tau(n^2) if n is odd, a(n) = tau(n^2) - a(n/2) if n is even.
%F Dirichlet g.f.: zeta^3(s)/(zeta(2s)*(1+1/2^s)). - _R. J. Mathar_, Jun 01 2011
%F Sum_{k=1..n} a(k) ~ 2*n / (9*Pi^2) * (9*log(n)^2 + 6*log(n) * (-3 + 9*g + log(2) - 36*Pi^(-2)*z1) + 18 + 54*g^2 + 18*g * (log(2) - 3) - 6*log(2) - log(2)^2 - 54*sg1 + 2592*z1^2/Pi^4 - 72*Pi^-2*(9*g*z1 + (log(2) - 3)*z1 + 3*z2)), where g is the Euler-Mascheroni constant A001620, sg1 is the first Stieltjes constant A082633, z1 = Zeta'(2) = A073002, z2 = Zeta''(2) = A201994. - _Vaclav Kotesovec_, Feb 02 2019
%F a(n) = Sum_{d|n} mu(2d)^2*tau(n/d), Dirichlet convolution of A323239 and A000005. - _Ridouane Oudra_, Nov 17 2019
%F a(n) = A361689(n)/2. - _R. J. Mathar_, Mar 21 2023
%p with(numtheory): seq(add(mobius(2*d)^2*tau(n/d), d in divisors(n)), n=1..100); # _Ridouane Oudra_, Nov 17 2019
%t Table[DivisorSigma[0, 2 n^2] / 2, {n, 100}] (* _Vincenzo Librandi_, Aug 14 2018 *)
%o (PARI) a(n) = numdiv(2*n^2)/2; \\ _Michel Marcus_, Oct 04 2013
%o (Sage) [sigma(2*n^2,0)/2 for n in range(1,100)] # _Joerg Arndt_, May 12 2014
%o (Magma) [NumberOfDivisors(2*n^2)/2 : n in [1..100]]; // _Vincenzo Librandi_, Aug 14 2018
%Y Cf. A000005, A001105, A048691.
%K mult,nonn,easy
%O 1,2
%A _Vladeta Jovovic_, Dec 13 2002
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