A099774 Number of divisors of 2*n-1.

1, 2, 2, 2, 3, 2, 2, 4, 2, 2, 4, 2, 3, 4, 2, 2, 4, 4, 2, 4, 2, 2, 6, 2, 3, 4, 2, 4, 4, 2, 2, 6, 4, 2, 4, 2, 2, 6, 4, 2, 5, 2, 4, 4, 2, 4, 4, 4, 2, 6, 2, 2, 8, 2, 2, 4, 2, 4, 6, 4, 3, 4, 4, 2, 4, 2, 4, 8, 2, 2, 4, 4, 4, 6, 2, 2, 6, 4, 2, 4, 4, 2, 8, 2, 3, 6, 2, 6, 4, 2, 2, 4, 4, 4, 8, 2, 2, 8, 2, 2, 4, 4, 4, 6, 4

Offset

1,2

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Formula

G.f.: Sum_{k>0} x^k/(1-x^(2*k-1)). - Michael Somos, Sep 02 2006

G.f.: sum(k>=1, x^((2*k-1)^2/2+1/2) * (1+x^(2*k-1))/(1-x^(2*k-1)) ). - Joerg Arndt, Nov 08 2010

Dirichlet g.f. (with interpolated zeros): zeta(s)^2*(1-1/2^s)^2. - Geoffrey Critzer, Feb 15 2015

Sum_{k=1..n} a(k) ~ (n*log(n) + (2*gamma - 1 + 3*log(2))*n)/2, where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 27 2022

Example

a(5)=3 because the divisors of 9 are: 1, 3 and 9.

Maple

with(numtheory): seq(tau(2*n-1), n=1..120);

Mathematica

nn = 200;
f[list_, i_] := list[[i]]; a =Table[Boole[OddQ[n]], {n, 1, nn}]; Select[Table[DirichletConvolve[f[a, n], f[a, n], n, m], {m, 1, nn}], # > 0 &] (* Geoffrey Critzer, Feb 15 2015 *)
Table[DivisorSigma[0, 2*n-1], {n, 1, 100}] (* Vaclav Kotesovec, Jan 14 2019 *)

Prog

(PARI) {a(n)=if(n<1, 0, numdiv(2*n-1))} /* Michael Somos, Sep 03 2006 */
(Haskell)
a099774 = a000005 . a005408  -- Reinhard Zumkeller, Sep 22 2014
(Magma) [NumberOfDivisors(2*n+1): n in [0..100]]; // Vincenzo Librandi, Mar 18 2015
(GAP) List([1..120], n->Tau(2*n-1)); # Muniru A Asiru, Dec 21 2018

Crossrefs

Bisection of A000005.

Cf. A000005, A001620, A099777.

Cf. A005408, A008438.

Sequence in context: A175778 A357039 A226182 * A305973 A290978 A142240

Adjacent sequences: A099771 A099772 A099773 * A099775 A099776 A099777

Keyword

nonn,easy

Author

N. J. A. Sloane, Nov 19 2004

Extensions

More terms from Emeric Deutsch, Dec 03 2004

Status

approved