A062011 - OEIS

%I #69 Oct 05 2022 04:59:40

%S 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,

%T 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,

%U 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

%N a(n) = 2*tau(n) = 2*A000005(n).

%C 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)."

%C 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

%C Number of divisors of p*n, where p is any prime not dividing n. - _Reinhard Zumkeller_, May 17 2006

%C From _Enrique Pérez Herrero_, Jul 21 2011: (Start)

%C 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.

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

%C Number of solutions to the Diophantine equation i*j = n*i + j. - _Robert G. Wilson v_, Apr 10 2019

%C 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

%H Harry J. Smith, <a href="/A062011/b062011.txt">Table of n, a(n) for n = 1..1000</a>

%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/poldiv12.jpg">Illustration of initial terms</a>

%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>

%F a(n) = A000005(A087560(n)) = A000005(A119416(n)). - _Reinhard Zumkeller_, May 17 2006

%F 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

%t A062011[n_] := 2*DivisorSigma[0,n]; Array[A062011,50] (* _Enrique Pérez Herrero_, Jul 15 2011 *)

%o (PARI) for (n=1, 1000, write("b062011.txt", n, " ", 2*numdiv(n))) \\ _Harry J. Smith_, Jul 29 2009

%Y Cf. A000005, A060648, A060710.

%Y Cf. A087560, A119416.

%Y Cf. A333119.

%K nonn,easy

%O 1,1

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

%E More terms from _Vladeta Jovovic_, Jul 14 2001

%E Edited by _N. J. A. Sloane_, Sep 20 2018, replacing old definition (which was of course correct) with a simple formula.