A162728 G.f.: x/(1-x) = Sum_{n>=1} a(n)*log(1+x^n)/n.

1, 3, 2, 8, 4, 6, 6, 20, 6, 12, 10, 16, 12, 18, 8, 48, 16, 18, 18, 32, 12, 30, 22, 40, 20, 36, 18, 48, 28, 24, 30, 112, 20, 48, 24, 48, 36, 54, 24, 80, 40, 36, 42, 80, 24, 66, 46, 96, 42, 60, 32, 96, 52, 54, 40, 120, 36, 84, 58, 64, 60, 90, 36, 256, 48, 60, 66, 128, 44, 72, 70

Offset

1,2

Comments

Dirichlet inverse of A117212. - R. J. Mathar, Jul 15 2010

Links

Paul D. Hanna, Table of n, a(n) for n = 1..10000

Formula

a(2n-1) = phi(2n-1); a(2n) = phi(2n)*A090739(n), where A090739(n) = exponent of 2 in 3^(2n)-1.

Inverse Mobius transform of A091512, where A091512(n) = exponent of 2 in (2n)^n.

Multiplicative: a(m,n) = a(m)*a(n) when gcd(m,n)=1, with a(p) = p-1 for odd prime p and a(2)=3.

G.f.: x/(1-x)^2 = Sum_{n>=1} a(n)*x^n/(1+x^n). - Paul D. Hanna, Jul 12 2009

Dirichlet g.f.: zeta(s-1)/( zeta(s)*(1-2^(1-s)) ). - R. J. Mathar, Apr 14 2011

a((2*n-1)*2^p) = (p+2)*2^(p-1)* phi(2*n-1), p >= 0. Observe that a(2^p) = A001792(p). - Johannes W. Meijer, Jan 26 2013

Sum_{k=1..n} a(k) ~ 6*n^2 / Pi^2. - Vaclav Kotesovec, Feb 07 2019

Multiplicative with a(2^e) = (e+2)*2^(e-1) and a(p^e) = (p-1)*p^(e-1) for an odd prime p. - Amiram Eldar, Aug 27 2023

Example

x/(1-x) = log(1+x) + 3*log(1+x^2)/2 + 2*log(1+x^3)/3 + 8*log(1+x^4)/4 + 4*log(1+x^5)/5 + 6*log(1+x^6)/6 + 6*log(1+x^7)/7 + 20*log(1+x^8)/8 +...

Maple

nmax:=71: with(numtheory): for p from 0 to ceil(simplify(log[2](nmax))) do for n from 1 to ceil(nmax/(p+2)) do a((2*n-1)*2^p) := (p+2)*2^(p-1)*phi(2*n-1) od: od: seq(a(n), n=1..nmax); # Johannes W. Meijer, Jan 26 2013

Mathematica

f[p_, e_] := (p-1)*p^(e-1); f[2, e_] := (e+2)*2^(e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 27 2023 *)

Prog

(PARI) /* As the inverse Mobius transform of A091512: */
{a(n)=sumdiv(n, d, moebius(n/d)*valuation((2*d)^d, 2))}
(PARI) /* From a(2n-1)=phi(2n-1); a(2n)=phi(2n)*A090739(n), we get: */
{a(n)=if(n%2==1, eulerphi(n), eulerphi(n)*valuation(3^n-1, 2))}
(PARI) /* From x/(1-x) = Sum_{n>=1} a(n)*log(1+x^n)/n, we get: */
{a(n)=local(A=[1]); for(k=1, n, A=concat(A, 0); A[ #A]=#A*(1-polcoeff(sum(m=1, #A, A[m]/m*log(1+x^m +x*O(x^#A)) ), #A))); A[n]}

Crossrefs

Cf. A090739, A091512, A000010 (Euler phi), A220466.

Sequence in context: A191731 A143515 A082333 * A127300 A129199 A211164

Adjacent sequences: A162725 A162726 A162727 * A162729 A162730 A162731

Keyword

mult,nonn,easy

Author

Paul D. Hanna, Jul 12 2009

Status

approved