A230283 -id:A230283 - cp's OEIS frontend

A230284 Denominators to Dirichlet inverse of Euler totient based version of series expansion for x/LambertW(x).

1, 1, 2, 3, 8, 15, 144, 35, 5760, 315, 5600, 693, 43545600, 1001, 6706022400, 6435, 14014, 109395, 376610217984000, 46189, 128047474114560000, 323323, 2540395, 2028117, 26976017466662584320000, 96577, 3241475864250624, 35102025, 2126818694000, 5386025

Offset: 1

Mats Granvik, Oct 15 2013

G. C. Greubel, Table of n, a(n) for n = 1..385

Cf. A191898, A177885, A230283 (numerators).

Similar to but strictly different from A264235.

Clear[nn, n, k, s, x]; nn = 22; Denominator[CoefficientList[1 + Integrate[1 + Expand[Sum[Exp[Limit[Zeta[s]*Sum[(If[n == 1, 0, Table[DivisorSum[m, # MoebiusMu[#] &], {m, nn}][[GCD[n, k]]]])/(k)^(s - 1), {k, 1, n}], s -> 1]]*(-x)^n, {n, 1, nn}]], x], x]]

A276048 Sequence associated with the functional equation of the Riemann zeta zero spectrum (see formulas).

Original entry on oeis.org

0, 2, 9, 2, 625, 1, 117649, 2, 9, 1, 25937424601, 1, 23298085122481, 1, 1, 2, 48661191875666868481, 1, 104127350297911241532841, 1, 1, 1, 907846434775996175406740561329, 1, 625, 1, 9, 1, 88540901833145211536614766025207452637361, 1

Offset: 1

Mats Granvik, Aug 17 2016

nonn

The functional equation formula in the answer by Peter Humphries is for the Dirichlet eta function and corresponds to the second term in this sequence. This sequence corresponds to zeta function products over all the divisors.

Peter Humphries, What completes the Dirichlet generating function ζ(s+c-1) where c is a constant?, Math StackExchange.
Peter Humphries, What is the symmetric functional equation of the Dirichlet eta function?, Math StackExchange.

Cf. A000169, A014963, A120112, A230283, A230284.

Clear[s]; -Table[Limit[Zeta[s]*Total[MoebiusMu[Divisors[n]]*Divisors[n]^(1 - (s))]*Total[MoebiusMu[Divisors[n]]*Divisors[n]^(s)], s -> 1], {n, 1, 30}]; Exp[%]

a(n) = exp(lim_{s->1} zeta(s)*Sum_{d|n} mu(d)*d^(1 - s)*Sum_{d|n} mu(d)*d^(s)).
a(n) = A014963(n)^(A014963(n)-1), n > 1.
a(n) = A014963(n)^(-A120112(n)), n > 1.
a(prime(n)) = A000169(prime(n)).

cp's OEIS Frontend

A230284 Denominators to Dirichlet inverse of Euler totient based version of series expansion for x/LambertW(x).

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