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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

S. Mustonen, Visualization and characterization of Pythagorean triples

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

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 xrange(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 19 05:08 EDT 2018. Contains 316336 sequences. (Running on oeis4.)