A057823 -id:A057823 - cp's OEIS frontend

A288968 Exponents a(1), a(2), ... such that E_2, 1 - 24q - 72q^2 - ... (A006352) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

24, 348, 6424, 129300, 2778648, 62114524, 1428337176, 33527349924, 799482197272, 19302454317660, 470740035601176, 11575875047000596, 286650683468840472, 7140515309818664028, 178783562850377621272, 4496350112540599930692

Offset: 1

Seiichi Manyama, Jun 20 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..701

Cf. this sequence (k=2), A110163 (k=4), A288851 (k=6), A288471 (k=8), A289024 (k=10), A288990/A288989 (k=12), A289029 (k=14).

Cf. A006352 (E_2), A008683, A288877 (E_4/E_2), A289635.

a(n) = 2 + (1/(12*n)) * Sum_{d|n} A008683(n/d) * A288877(d).
a(n) = (1/n) * Sum_{d|n} A008683(n/d) * A289635(d).
a(n) ~ 1 / (n * r^(2*n)), where r = A057823. - Vaclav Kotesovec, Mar 08 2018

A211342 Decimal expansion of q between 0 and 1 maximizing Dedekind eta function eta(q) = q^(1/24) * Product_{n>=1} (1 - q^n).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Feb 05 2013

			0.0372768102964516581509807856516446180362823794827830067...

Eric Weisstein's MathWorld, Dedekind Eta Function

Cf. A000203, A057823, A288877, A289392.

q /. Last @ FindMaximum[ DedekindEta[ -I*Log[q]/(2*Pi)], {q, 1/25}, WorkingPrecision -> 200] // RealDigits[#][[1]][[1 ;; 100]]& // Prepend[#, 0]&
x /. FindRoot[24*Sum[DivisorSigma[1, k]*x^k, {k, 1, 1000}] == 1, {x, 1}, WorkingPrecision -> 101] (* Vaclav Kotesovec, Jun 28 2017 *)

Root of the equation Sum_{k>=1} A000203(k) * r^k = 1/24. - Vaclav Kotesovec, Jun 28 2017

Equals A057823^2. - Vaclav Kotesovec, Jul 02 2017

cp's OEIS Frontend

A288968 Exponents a(1), a(2), ... such that E_2, 1 - 24q - 72q^2 - ... (A006352) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

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A211342 Decimal expansion of q between 0 and 1 maximizing Dedekind eta function eta(q) = q^(1/24) * Product_{n>=1} (1 - q^n).

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cp's OEIS Frontend

A288968 Exponents a(1), a(2), ... such that E_2, 1 - 24*q - 72*q^2 - ... (A006352) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

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A211342 Decimal expansion of q between 0 and 1 maximizing Dedekind eta function eta(q) = q^(1/24) * Product_{n>=1} (1 - q^n).

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Formula

A288968 Exponents a(1), a(2), ... such that E_2, 1 - 24q - 72q^2 - ... (A006352) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .