|
%I #53 Dec 26 2022 09:45:17
%S 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,
%T 28,12,30,16,30,16,24,18,36,18,36,16,40,18,42,20,36,22,46,24,42,20,48,
%U 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
%N a(n) = phi(3*n)/2.
%C Compare the o.g.f. of this sequence to the following identity:
%C Sum_{n>=1} -moebius(3*n)*x^n/(1-x^n) = Sum_{n>=0} x^(3^n).
%C Here phi(n) = A000010(n), the Euler totient function of n.
%C 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
%H Paul D. Hanna, <a href="/A195459/b195459.txt">Table of n, a(n) for n = 1..1000</a>
%H M. Picquet, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k433703n/f35.image">Applications de la représentation des courbes du troisième degré</a>, Journal de l'École Polytechnique, Paris, 35 (1884), pp. 31-100.
%F O.g.f.: Sum_{n>=1} -moebius(3*n)*x^n/(1-x^n)^2 = Sum_{n>=1} phi(3*n)/2*x^n.
%F a(n) = n * Product_{p | n, p prime, p != 3} (1 - 1/p). [Picquet, p. 73.]
%F a(n) = phi(n)/2*(((2*n^2+1) mod 3)+2). - _Gary Detlefs_, Dec 06 2021
%F Sum_{k=1..n} a(k) ~ c * n^2, where c = 27/(8*Pi^2) = 0.341958... . - _Amiram Eldar_, Nov 18 2022
%F Dirichlet g.f.: zeta(s-1)/(zeta(s)*(1-1/3^s)). - _Amiram Eldar_, Dec 26 2022
%e 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 +...
%e 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 +...
%t EulerPhi[3*Range[100]]/2 (* _Harvey P. Dale_, Mar 08 2022 *)
%o (PARI) {a(n)=polcoeff(sum(m=1, n, -moebius(3*m)*x^m/(1-x^m+x*O(x^n))^2), n)}
%o (PARI) a(n) = eulerphi(3*n)/2; \\ _Michel Marcus_, Jun 07 2020
%Y Cf. A000010 (phi), A023022 (phi/2), A062570.
%K nonn,mult
%O 1,3
%A _Paul D. Hanna_, Sep 18 2011
%E Picquet formula and reference added by _N. J. A. Sloane_, Nov 23 2011
|
