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A123726
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Denominators of fractional partial quotients appearing in a continued fraction for the power series Sum_{n>=0} x^(2^n - 1)/(n+1)^s.
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2
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1, 1, 4, 1, 9, 1, 4, 1, 16, 1, 4, 1, 9, 1, 4, 1, 25, 1, 4, 1, 9, 1, 4, 1, 16, 1, 4, 1, 9, 1, 4, 1, 36, 1, 4, 1, 9, 1, 4, 1, 16, 1, 4, 1, 9, 1, 4, 1, 25, 1, 4, 1, 9, 1, 4, 1, 16, 1, 4, 1, 9, 1, 4, 1, 49, 1, 4, 1, 9, 1, 4, 1, 16, 1, 4, 1, 9, 1, 4, 1, 25, 1, 4, 1, 9, 1, 4, 1, 16, 1, 4, 1, 9, 1, 4, 1, 36, 1, 4
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OFFSET
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0,3
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COMMENTS
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LINKS
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FORMULA
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a(n) = (A007814(n) + 1)^2 = A001511(n)^2 for n>=1, with a(0)=1, where A007814(n) is the exponent of the highest power of 2 dividing n.
Multiplicative with a(2^e) = (e + 1)^2, a(p^e) = 1 for odd prime p. - Andrew Howroyd, Jul 31 2018
Dirichlet g.f.: zeta(s)*(4^s+2^s)/(2^s-1)^2.
Sum_{k=1..n} a(k) ~ 6*n. (End)
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EXAMPLE
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Surprisingly, the following analog of the Riemann zeta function:
Z(x,s) = Sum_{n>=0} x^(2^n-1)/(n+1)^s = 1 + x/2^s + x^3/3^s +x^7/4^s+..
may be expressed by the continued fraction:
Z(x,s) = [1; f(1)^s/x, -f(2)^s/x, -f(3)^s/x,...,f(n)^s*(-1)^e(n)/x,...]
such that the (2^n-1)-th convergent = Sum_{k=0..n} x^(2^k-1)/(k+1)^s,
where f(n) = (b(n)+2)/(b(n)+1)^2 and e(n) = A073089(n+1) for n>=1,
and b(n) = A007814(n) the exponent of highest power of 2 dividing n.
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MATHEMATICA
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Join[{1}, Table[(1 + IntegerExponent[n, 2])^2, {n, 1, 50}]] (* G. C. Greubel, Nov 01 2018 *)
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PROG
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(PARI) {a(n)=denominator(subst(contfrac(sum(m=0, #binary(n), 1/x^(2^m-1)/(m+1)), n+3)[n+1], x, 1))}
(PARI) /* a(n) = (A007814(n)+1)^2: */ {a(n)=if(n==0, 1, (valuation(n, 2)+1)^2)}
(Magma) [1] cat [(Valuation(n, 2) + 1)^2: n in [1..50]]; // G. C. Greubel, Nov 01 2018
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CROSSREFS
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KEYWORD
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frac,nonn,mult
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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