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A162728
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G.f.: x/(1-x) = Sum_{n>=1} a(n)*log(1+x^n)/n.
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1
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Dirichlet inverse of A117212. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 15 2010]
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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). [From Paul D. Hanna (pauldhanna(AT)juno.com), Jul 12 2009]
Dirichlet g.f. zeta(s-1)/( zeta(s)*(1-2^(1-s)) ). - R. J. Mathar, Apr 14 2011
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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 +...
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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]}
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CROSSREFS
| Cf. A090739, A091512, A000010 (Euler phi).
Sequence in context: A191731 A143515 A082333 * A127300 A129199 A097018
Adjacent sequences: A162725 A162726 A162727 * A162729 A162730 A162731
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KEYWORD
| mult,nonn
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Jul 12 2009
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