A345327 Decimal expansion of a constant Y2 related to the asymptotics of A000203.

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

Offset

0,1

Links

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

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 51 (constant Y2).

V. Sita Ramaiah and D. Suryanarayana, Sums of reciprocals of some multiplicative functions, Mathematical Journal of Okayama University, Vol. 21, No. 2 (1979), pp. 161-162, formula (4.1)-(4.4).

László Tóth, Alternating sums concerning multiplicative arithmetic functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1 (section 4.3, p. 18).

Formula

Equals Sum_{p primes} (p-1)^2 * g(p) * log(p) / (p*f(p)), where f(p) = 1 - (p-1)^2/p * Sum_{j>=1} 1/((p^j - 1)*(p^(j+1) - 1)) and g(p) = Sum_{j>=1} j/((p^j - 1)*(p^(j+1) - 1)).

Sum_{k=1..n} 1/A000203(k) ~ Y1*log(n) + Y1*(gamma + Y2), where gamma = A001620, Y1 = A308039, Y2 = A345327.

Example

0.5073388825830843781004978765159526773890196348281644808049...

Mathematica

$MaxExtraPrecision = 1000; Do[ratfun = (p - 1)^2 * Sum[j/(p^j - 1)/(p^(j + 1) - 1), {j, 1, m}]/(p*(1 - (p - 1)^2/p * Sum[1/(p^j - 1)/(p^(j + 1) - 1), {j, 1, m}])); zetas = 0; ratab = Table[konfun = Together[ratfun + c/(p^power - 1)]; 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, 40}]; Print[N[Sum[Log[p]*ratfun /. p -> Prime[k], {k, 1, m}] + zetas, 110]], {m, 10, 250, 10}]

Crossrefs

Cf. A000203, A308039.

Sequence in context: A201656 A168195 A333507 * A322712 A244345 A021873

Adjacent sequences: A345324 A345325 A345326 * A345328 A345329 A345330

Keyword

nonn,cons

Author

Vaclav Kotesovec, Jun 14 2021

Status

approved