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A345327
Decimal expansion of a constant Y2 related to the asymptotics of A000203.
2
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
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
Sequence in context: A201656 A168195 A333507 * A322712 A244345 A021873
KEYWORD
nonn,cons
AUTHOR
Vaclav Kotesovec, Jun 14 2021
STATUS
approved