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A161166 Decimal expansion of a constant in the linear term in the growth rate of unitary squarefree divisors. 6

%I

%S 7,4,8,3,7,2,3,3,3,4,2,9,6,7,4,7,0,0,9,3,8,0,8,6,5,2,9,4,3,9,4,0,8,9,

%T 9,5,9,9,2,9,2,5,4,0,2,5,9,4,5,6,8,9,6,6,0,0,0,8,5,5,1,3,0,8,8,5,7,5,

%U 2,5,6,7,6,9,7,5,1,3,0,8,3,9,6,4,5,9,3,8,4,2,6,2,1,1,9,7,1,0,0,8,1,5,5,6,8,2

%N Decimal expansion of a constant in the linear term in the growth rate of unitary squarefree divisors.

%C Other constituents of the linear term are in A065463, A073002, A001620 and A059956.

%D D. Suryanarayana and V. Siva Rama Prasad, The number of k-ary, k+1-free divisors of an integer, J. Reine Angew. Math. 276 (1975) 15-35.

%H Steven R. Finch, <a href="/A007947/a007947.pdf">Unitarism and Infinitarism</a>, February 25, 2004. [Cached copy, with permission of the author]

%F Equals sum_{primes p} (2p+1)*log(p)/((p+1)(p^2+p-1)) = sum_p log(p)*[2/(p^2-1)-3/p^3-1)+4/(p^4-1)-10/(p^5-1)....] where the terms accumulate; this is essentially the logarithmic derivative of the Riemann zeta function at s=2, 3, 4,...

%e 0.748372333429674...

%t ratfun = (2*p + 1)/((p + 1)*(p^2 + p - 1)); zetas = 0; ratab = Table[konfun = Simplify[ratfun + c/(p^power - 1)] // Together; coefs = CoefficientList[Numerator[konfun], p]; sol = Solve[Last[coefs] == 0, c][[1]]; zetas = zetas + c*Zeta'[power]/Zeta[power] /. sol; ratfun = konfun /. sol, {power, 2, 25}]; Do[Print[N[Sum[Log[p]*ratfun /. p -> Prime[k], {k, 1, m}] + zetas, 120]], {m, 2000, 20000, 2000}] (* _Vaclav Kotesovec_, Jun 24 2020 *)

%K cons,nonn

%O 0,1

%A _R. J. Mathar_, Jun 04 2009

%E More digits from _Vaclav Kotesovec_, Jun 24 2020

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Last modified July 8 05:33 EDT 2020. Contains 335513 sequences. (Running on oeis4.)