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
Examples
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 +...
Links
- Paul D. Hanna, Table of n, a(n) for n = 1..10000
Crossrefs
Programs
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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
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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 *)
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PARI
/* As the inverse Mobius transform of A091512: */ {a(n)=sumdiv(n,d,moebius(n/d)*valuation((2*d)^d,2))}
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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))}
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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]}
Formula
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
From Ridouane Oudra, Jul 05 2025: (Start)
a(n) = Sum_{k=0..A007814(n)} 2^k*phi(n/2^k).
a(n) = Sum_{d|n} mu(n/d)*d*A001511(d).
Comments