

A078644


a(n) = tau(2*n^2)/2.


5



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
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



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


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



PROG

(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



KEYWORD

mult,nonn,easy


AUTHOR



STATUS

approved



