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 A078644 a(n) = tau(2*n^2)/2. 4
 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, 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, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Inverse Moebius transform of A068068. Number of elements in the set {(x,y): x is odd, x|n, y|n, gcd(x,y)=1}. 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 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 Dirichlet convolution of A048691 and A154269. - R. J. Mathar, Jun 01 2011 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 LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..10000 Seppo Mustonen, Visualization and characterization of Pythagorean triples [Local copy] T. Omland, How many Pythagorean triples with given inradius?, J. Numb. Theory 170 (2017) 1-2. FORMULA 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. Dirichlet g.f.: zeta^3(s)/(zeta(2s)*(1+1/2^s)). - R. J. Mathar, Jun 01 2011 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 a(n) = Sum_{d|n} mu(2d)^2*tau(n/d). - Ridouane Oudra, Nov 17 2019 MAPLE with(numtheory): seq(add(mobius(2*d)^2*tau(n/d), d in divisors(n)), n=1..100); # Ridouane Oudra, Nov 17 2019 MATHEMATICA Table[DivisorSigma[0, 2 n^2] / 2, {n, 100}] (* Vincenzo Librandi, Aug 14 2018 *) PROG (PARI) a(n) = numdiv(2*n^2)/2; \\ Michel Marcus, Oct 04 2013 (Sage)  [sigma(2*n^2, 0)/2 for n in range(1, 100)] # Joerg Arndt, May 12 2014 (Magma) [NumberOfDivisors(2*n^2)/2 : n in [1..100]]; // Vincenzo Librandi, Aug 14 2018 CROSSREFS Cf. A000005, A001105, A048691. Sequence in context: A119688 A126868 A134187 * A133700 A087688 A126854 Adjacent sequences:  A078641 A078642 A078643 * A078645 A078646 A078647 KEYWORD mult,nonn,easy AUTHOR Vladeta Jovovic, Dec 13 2002 STATUS approved

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Last modified October 7 05:10 EDT 2022. Contains 357270 sequences. (Running on oeis4.)