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

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, 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, 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

Offset

0,1

Comments

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

References

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.

Links

Table of n, a(n) for n=0..105.

Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 50.

Formula

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,...

Example

0.748372333429674...

Mathematica

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 *)

Crossrefs

Sequence in context: A021576 A222183 A010509 * A199060 A330596 A296427

Adjacent sequences: A161163 A161164 A161165 * A161167 A161168 A161169

Keyword

cons,nonn

Author

R. J. Mathar, Jun 04 2009

Extensions

More digits from Vaclav Kotesovec, Jun 24 2020

Status

approved