A062011 a(n) = 2*tau(n) = 2*A000005(n).

2, 4, 4, 6, 4, 8, 4, 8, 6, 8, 4, 12, 4, 8, 8, 10, 4, 12, 4, 12, 8, 8, 4, 16, 6, 8, 8, 12, 4, 16, 4, 12, 8, 8, 8, 18, 4, 8, 8, 16, 4, 16, 4, 12, 12, 8, 4, 20, 6, 12, 8, 12, 4, 16, 8, 16, 8, 8, 4, 24, 4, 8, 12, 14, 8, 16, 4, 12, 8, 16, 4, 24, 4, 8, 12, 12, 8, 16, 4, 20, 10, 8, 4, 24, 8, 8, 8, 16

Offset: 1

Author

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 12 2001

Keywords

• nonn
• easy

Comments

Old definition was "Number of cyclic subgroups of the group C_n X C_2 (where C_n is the cyclic group with n elements)."

More generally, the number of cyclic subgroups of the group C_n X C_m is Sum_{i|n, j|m} phi(i)*phi(j)/phi(lcm(i,j)), where phi=Euler totient function, cf. A000010. - Vladeta Jovovic, Jul 15 2001

Number of divisors of p*n, where p is any prime not dividing n. - Reinhard Zumkeller, May 17 2006

From Enrique Pérez Herrero, Jul 21 2011: (Start)

If p(x) is a polynomial with integer coefficients, and if r is an integer zero of p(x), then r is a divisor of the constant term c_0 of p(x). Under this theorem, p(x) can have a(c_0) possible integer roots.

a(n) is the number of integer divisors of n, while A000005(n) is the number of positive divisors. (End)

Number of solutions to the Diophantine equation i*j = n*i + j. - Robert G. Wilson v, Apr 10 2019

a(n) is also the number of times n appears in the triangle A333119, or equivalently, the number of positive integer solutions of the equation A333119(x, y) = n for y < x. - Stefano Spezia, Oct 05 2022

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1000
• Omar E. Pol, Illustration of initial terms
• Index entries for sequences related to groups

Crossrefs

Cf. A000005, A060648, A060710.

Cf. A087560, A119416.

Cf. A333119.

Programs

Mathematica

A062011[n_] := 2*DivisorSigma[0,n]; Array[A062011,50] (* Enrique Pérez Herrero, Jul 15 2011 *)

PARI

for (n=1, 1000, write("b062011.txt", n, " ", 2*numdiv(n))) \\ Harry J. Smith, Jul 29 2009

Formula

a(n) = A000005(A087560(n)) = A000005(A119416(n)). - Reinhard Zumkeller, May 17 2006

L.g.f.: -log(Product_{k>=1} (1 - x^k)^(2/k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Mar 18 2018

Extensions

More terms from Vladeta Jovovic, Jul 14 2001

Edited by N. J. A. Sloane, Sep 20 2018, replacing old definition (which was of course correct) with a simple formula.