

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
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OFFSET

1,2


COMMENTS

Inverse Moebius transform of A068068. Number of elements in the set {(x,y): x is odd, xn, yn, gcd(x,y)=1}.
The number of Pythagorean points (x,y), 0 < x < y, located on the hyperbola y = 2n(xn)/(x2n) and having "excess" x+yz = 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 Lshapes 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 Lshapes. So Lshapes have formula: 2xy = (x+n)(y+n).  Juhani Heino, Jul 23 2012


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
S. Mustonen, Visualization and characterization of Pythagorean triples
Seppo Mustonen, Visualization and characterization of Pythagorean triples [Local copy]
T. Omland, How many Pythagorean triples with given inradius?, J. Numb. Theory 170 (2017) 12.


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 EulerMascheroni 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_{dn} 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



