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A123726 Denominators of fractional partial quotients appearing in a continued fraction for the power series Sum_{n>=0} x^(2^n - 1)/(n+1)^s.

%I A123726 #22 Dec 29 2022 06:31:14
%S A123726 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,
%T A123726 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,
%U A123726 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
%N A123726 Denominators of fractional partial quotients appearing in a continued fraction for the power series Sum_{n>=0} x^(2^n - 1)/(n+1)^s.
%C A123726 A123725(n) = (A007814(n) + 2)*(-1)^A073089(n+1) are the numerators of the partial quotients.
%H A123726 Antti Karttunen, <a href="/A123726/b123726.txt">Table of n, a(n) for n = 0..65537</a>
%F A123726 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.
%F A123726 Multiplicative with a(2^e) = (e + 1)^2, a(p^e) = 1 for odd prime p. - _Andrew Howroyd_, Jul 31 2018
%F A123726 From _Amiram Eldar_, Dec 29 2022: (Start)
%F A123726 Dirichlet g.f.: zeta(s)*(4^s+2^s)/(2^s-1)^2.
%F A123726 Sum_{k=1..n} a(k) ~ 6*n. (End)
%e A123726 Surprisingly, the following analog of the Riemann zeta function:
%e A123726 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+..
%e A123726 may be expressed by the continued fraction:
%e A123726 Z(x,s) = [1; f(1)^s/x, -f(2)^s/x, -f(3)^s/x,...,f(n)^s*(-1)^e(n)/x,...]
%e A123726 such that the (2^n-1)-th convergent = Sum_{k=0..n} x^(2^k-1)/(k+1)^s,
%e A123726 where f(n) = (b(n)+2)/(b(n)+1)^2 and e(n) = A073089(n+1) for n>=1,
%e A123726 and b(n) = A007814(n) the exponent of highest power of 2 dividing n.
%t A123726 Join[{1}, Table[(1 + IntegerExponent[n, 2])^2, {n, 1, 50}]] (* _G. C. Greubel_, Nov 01 2018 *)
%o A123726 (PARI) {a(n)=denominator(subst(contfrac(sum(m=0,#binary(n),1/x^(2^m-1)/(m+1)),n+3)[n+1],x,1))}
%o A123726 (PARI) /* a(n) = (A007814(n)+1)^2: */ {a(n)=if(n==0,1,(valuation(n,2)+1)^2)}
%o A123726 (Magma) [1] cat [(Valuation(n, 2) + 1)^2: n in [1..50]]; // _G. C. Greubel_, Nov 01 2018
%Y A123726 Cf. A123725 (numerators), A007814, A073089, A001511.
%K A123726 frac,nonn,mult
%O A123726 0,3
%A A123726 _Paul D. Hanna_, Oct 12 2006
%E A123726 Ref to A001511 added by _Franklin T. Adams-Watters_, Dec 22 2013