A195459 a(n) = phi(3*n)/2.

1, 1, 3, 2, 4, 3, 6, 4, 9, 4, 10, 6, 12, 6, 12, 8, 16, 9, 18, 8, 18, 10, 22, 12, 20, 12, 27, 12, 28, 12, 30, 16, 30, 16, 24, 18, 36, 18, 36, 16, 40, 18, 42, 20, 36, 22, 46, 24, 42, 20, 48, 24, 52, 27, 40, 24, 54, 28, 58, 24, 60, 30, 54, 32, 48, 30, 66, 32, 66, 24, 70, 36, 72, 36, 60, 36, 60, 36, 78, 32, 81, 40, 82

Offset: 1

Author

Paul D. Hanna, Sep 18 2011

Keywords

• nonn
• mult

Comments

Compare the o.g.f. of this sequence to the following identity:

Sum_{n>=1} -moebius(3*n)*x^n/(1-x^n) = Sum_{n>=0} x^(3^n).

Here phi(n) = A000010(n), the Euler totient function of n.

a(n) = b(n)*c(n) where b(n) = 1, 1, 3, 2, 1,.. is a multiplicative function with b(p^e) = p^(e-1) for p<>3 and p(3^e)=3^e, and where c(n) = 1, 1, 1, 1, 4, 1, 6, 1, 1... is a multiplicative function with c(p^e)=p-1 for p <> 3 and c(3^e)=1. - R. J. Mathar, Jul 02 2013

Examples

			G.f.: A(x) = x + x^2 + 3*x^3 + 2*x^4 + 4*x^5 + 3*x^6 + 6*x^7 + 4*x^8 +...
where A(x) = x/(1-x)^2 - x^2/(1-x^2)^2 + 0*x^3/(1-x^3)^2 + 0*x^4/(1-x^4)^2 - x^5/(1-x^5)^2 + 0*x^6/(1-x^6)^2 - x^7/(1-x^7)^2 + 0*x^8/(1-x^8)^2 + 0*x^9/(1-x^9)^2 + x^10/(1-x^10)^2 - x^11/(1-x^11)^2 +...+ -moebius(3*n)*x^n/(1-x^n)^2 +...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..1000
• M. Picquet, Applications de la représentation des courbes du troisième degré, Journal de l'École Polytechnique, Paris, 35 (1884), pp. 31-100.

Crossrefs

Cf. A000010 (phi), A023022 (phi/2), A062570.

Programs

Mathematica

EulerPhi[3*Range[100]]/2 (* Harvey P. Dale, Mar 08 2022 *)

PARI

{a(n)=polcoeff(sum(m=1, n, -moebius(3*m)*x^m/(1-x^m+x*O(x^n))^2), n)}

PARI

a(n) = eulerphi(3*n)/2; \\ Michel Marcus, Jun 07 2020

Formula

O.g.f.: Sum_{n>=1} -moebius(3*n)*x^n/(1-x^n)^2 = Sum_{n>=1} phi(3*n)/2*x^n.

a(n) = n * Product_{p | n, p prime, p != 3} (1 - 1/p). [Picquet, p. 73.]

a(n) = phi(n)/2*(((2*n^2+1) mod 3)+2). - Gary Detlefs, Dec 06 2021

Sum_{k=1..n} a(k) ~ c * n^2, where c = 27/(8*Pi^2) = 0.341958... . - Amiram Eldar, Nov 18 2022

Dirichlet g.f.: zeta(s-1)/(zeta(s)*(1-1/3^s)). - Amiram Eldar, Dec 26 2022

Extensions

Picquet formula and reference added by N. J. A. Sloane, Nov 23 2011