A292888 -id:A292888 - cp's OEIS frontend

A259149 Decimal expansion of phi(exp(-2*Pi)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.

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

Offset: 0

Jean-François Alcover, Jun 19 2015

			0.99812906992595851327996232224527387813073843536581646175407814028299858...

Istvan Mezo, Several special values of Jacobi theta functions arXiv:1106.2703 [math.CA], 2011-2013
Eric Weisstein's MathWorld, Infinite Product
Eric Weisstein's MathWorld, Jacobi Theta Functions
Eric Weisstein's MathWorld, q-Pochhammer Symbol
Wikipedia, Euler function

Cf. A048651 phi(1/2), A100220 phi(1/3), A100221 phi(1/4), A100222 phi(1/5), A132034 phi(1/6), A132035 phi(1/7), A132036 phi(1/8), A132037 phi(1/9), A132038 phi(1/10), A368211 phi(exp(-Pi/16)), A292862 phi(exp(-Pi/8)), A292863 phi(exp(-Pi/4)), A259147 phi(exp(-Pi/2)), A259148 phi(exp(-Pi)), A292888 phi(exp(-3*Pi)), A259150 phi(exp(-4*Pi)), A292905 phi(exp(-5*Pi)), A363018 phi(exp(-6*Pi)), A363117 phi(exp(-7*Pi)), A259151 phi(exp(-8*Pi)), A363118 phi(exp(-9*Pi)), A363019 phi(exp(-10*Pi)), A363081 phi(exp(-11*Pi)), A363020 phi(exp(-12*Pi)), A363178 phi(exp(-13*Pi)), A363119 phi(exp(-14*Pi)), A363179 phi(exp(-15*Pi)), A292864 phi(exp(-16*Pi)), A363120 phi(exp(-18*Pi)), A363021 phi(exp(-20*Pi)).

Cf. A292821, A292825, A292829.

phi[q_] := QPochhammer[q, q]; RealDigits[phi[Exp[-2Pi]], 10, 104] // First

phi(q) = QPochhammer(q,q) = (q;q)_infinity.
phi(q) also equals theta'(1, 0, sqrt(q))^(1/3)/(2^(1/3)*q^(1/24)), where theta' is the derivative of the elliptic theta function theta(a,u,q) w.r.t. u.
phi(exp(-2*Pi)) = exp(Pi/12)*Gamma(1/4)/(2*Pi^(3/4)).

A259148 Decimal expansion of phi(exp(-Pi)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 19 2015

			0.954918789987674103751233978110291077632715373807805283148799191676094...

Istvan Mezo, Several special values of Jacobi theta functions, arXiv:1106.2703 [math.CA], 2011-2013.
Eric Weisstein's MathWorld, Infinite Product
Eric Weisstein's MathWorld, Jacobi Theta Functions
Eric Weisstein's MathWorld, q-Pochhammer Symbol
Wikipedia, Euler function

Cf. A048651 phi(1/2), A100220 phi(1/3), A100221 phi(1/4), A100222 phi(1/5), A132034 phi(1/6), A132035 phi(1/7), A132036 phi(1/8), A132037 phi(1/9), A132038 phi(1/10), A368211 phi(exp(-Pi/16)), A292862 phi(exp(-Pi/8)), A292863 phi(exp(-Pi/4)), A259147 phi(exp(-Pi/2)), A259149 phi(exp(-2*Pi)), A292888 phi(exp(-3*Pi)), A259150 phi(exp(-4*Pi)), A292905 phi(exp(-5*Pi)), A363018 phi(exp(-6*Pi)), A363117 phi(exp(-7*Pi)), A259151 phi(exp(-8*Pi)), A363118 phi(exp(-9*Pi)), A363019 phi(exp(-10*Pi)), A363081 phi(exp(-11*Pi)), A363020 phi(exp(-12*Pi)), A363178 phi(exp(-13*Pi)), A363119 phi(exp(-14*Pi)), A363179 phi(exp(-15*Pi)), A292864 phi(exp(-16*Pi)), A363120 phi(exp(-18*Pi)), A363021 phi(exp(-20*Pi)).

Cf. A292820, A292824, A292828.

phi[q_] := QPochhammer[q, q]; RealDigits[phi[Exp[-Pi]], 10, 104] // First

phi(q) = QPochhammer(q,q) = (q;q)_infinity.
phi(q) also equals theta'(1, 0, sqrt(q))^(1/3)/(2^(1/3)*q^(1/24)), where theta' is the derivative of the elliptic theta function theta(a,u,q) w.r.t. u.
phi(exp(-Pi)) = exp(Pi/24)*Gamma(1/4)/(2^(7/8)*Pi^(3/4)).
Equals 1/exp(A255695). - Hugo Pfoertner, May 28 2025

A259150 Decimal expansion of phi(exp(-4*Pi)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 19 2015

			0.99999651264548223429509891685211924765750978932634584844773269100472...

Istvan Mezo, Several special values of Jacobi theta functions, arXiv:1106.2703 [math.CA], 2011-2013.
Eric Weisstein's MathWorld, Infinite Product
Eric Weisstein's MathWorld, Jacobi Theta Functions
Eric Weisstein's MathWorld, q-Pochhammer Symbol
Wikipedia, Euler function

Cf. A048651 phi(1/2), A100220 phi(1/3), A100221 phi(1/4), A100222 phi(1/5), A132034 phi(1/6), A132035 phi(1/7), A132036 phi(1/8), A132037 phi(1/9), A132038 phi(1/10), A292862 phi(exp(-Pi/8)), A292863 phi(exp(-Pi/4)), A259147 phi(exp(-Pi/2)), A259148 phi(exp(-Pi)), A259149 phi(exp(-2*Pi)), A292888 phi(exp(-3*Pi)), A292905 phi(exp(-5*Pi)), A363018 phi(exp(-6*Pi)), A363117 phi(exp(-7*Pi)), A259151 phi(exp(-8*Pi)), A363118 phi(exp(-9*Pi)), A363019 phi(exp(-10*Pi)), A363081 phi(exp(-11*Pi)), A363020 phi(exp(-12*Pi)), A363178 phi(exp(-13*Pi)), A363119 phi(exp(-14*Pi)), A363179 phi(exp(-15*Pi)), A292864 phi(exp(-16*Pi)), A363120 phi(exp(-18*Pi)), A363021 phi(exp(-20*Pi)).

Cf. A000706, A292822, A292826.

phi[q_] := QPochhammer[q, q]; RealDigits[phi[Exp[-4*Pi]], 10, 103] // First

phi(q) = QPochhammer(q,q) = (q;q)_infinity.
phi(q) also equals theta'(1, 0, sqrt(q))^(1/3)/(2^(1/3)*q^(1/24)), where theta' is the derivative of the elliptic theta function theta(a,u,q) w.r.t. u.
phi(exp(-4*Pi)) = exp(Pi/6)*Gamma(1/4)/(2^(11/8)*Pi^(3/4)).
A259150 = A259148 * exp(Pi/8)/sqrt(2). - Vaclav Kotesovec, Jul 03 2017

A259151 Decimal expansion of phi(exp(-8*Pi)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 19 2015

			0.999999999987838443290442788140998270959486945673852198543872725583699...

Istvan Mezo, Several special values of Jacobi theta functions, arXiv:1106.2703 [math.CA], 2011-2013.
Eric Weisstein's MathWorld, Infinite Product
Eric Weisstein's MathWorld, Jacobi Theta Functions
Eric Weisstein's MathWorld, q-Pochhammer Symbol
Wikipedia, Euler function

Cf. A048651 phi(1/2), A100220 phi(1/3), A100221 phi(1/4), A100222 phi(1/5), A132034 phi(1/6), A132035 phi(1/7), A132036 phi(1/8), A132037 phi(1/9), A132038 phi(1/10), A292862 phi(exp(-Pi/8)), A292863 phi(exp(-Pi/4)), A259147 phi(exp(-Pi/2)), A259148 phi(exp(-Pi)), A259149 phi(exp(-2*Pi)), A292888 phi(exp(-3*Pi)), A259150 phi(exp(-4*Pi)), A292905 phi(exp(-5*Pi)), A363018 phi(exp(-6*Pi)), A363117 phi(exp(-7*Pi)), A363118 phi(exp(-9*Pi)), A363019 phi(exp(-10*Pi)), A363081 phi(exp(-11*Pi)), A363020 phi(exp(-12*Pi)), A363178 phi(exp(-13*Pi)), A363119 phi(exp(-14*Pi)), A363179 phi(exp(-15*Pi)), A292864 phi(exp(-16*Pi)), A363120 phi(exp(-18*Pi)), A363021 phi(exp(-20*Pi)).

phi[q_] := QPochhammer[q, q]; RealDigits[phi[Exp[-8*Pi]], 10, 103] // First

phi(q) = QPochhammer(q,q) = (q;q)_infinity.
phi(q) also equals theta'(1, 0, sqrt(q))^(1/3)/(2^(1/3)*q^(1/24)), where theta' is the derivative of the elliptic theta function theta(a,u,q) w.r.t. u.
phi(exp(-8*Pi)) = (sqrt(2) - 1)^(1/4)*exp(Pi/3)*(Gamma(1/4)/(2^(29/16)*Pi^(3/4))).
A259151 = A259147 * exp(5*Pi/16)/2. - Vaclav Kotesovec, Jul 03 2017

A292905 Decimal expansion of Product_{k>=1} (1 - exp(-5Pik)).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Sep 26 2017

			0.999999849298249749982855684249951337192226280495972174466518680326272...

Eric Weisstein's World of Mathematics, Dedekind Eta Function
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol
Wikipedia, Dedekind eta function
Wikipedia, Euler function

Cf. A292862, A292863, A259147, A259148, A259149, A292888, A259150, A259151, A292864.

Cf. A292904.

RealDigits[QPochhammer[E^(-5*Pi)], 10, 120][[1]]
RealDigits[E^(5*Pi/8) * Gamma[1/4] * (9 + 4*Sqrt[5])^(1/4) * (-E^(5*Pi/2) + Sqrt[E^(5*Pi) + 64*r^24])^(1/4) / (2^(7/4) * Sqrt[5] * Pi^(3/4) * r^5)/.r -> (r/.FindRoot[2^(3/4)*r^6 + 2^(17/8)*E^(5*Pi/24)*r^5 + 2^(5/8)*E^(25*Pi/24)*r - E^(5*Pi/4) == 0, {r, 1}, WorkingPrecision -> 130]), 10, 120][[1]]
RealDigits[E^(5*Pi/24) * Gamma[1/4]*(7 + 3*Sqrt[5] + 12*Sqrt[14*Sqrt[5] - 30])^(1/8) / (2*Sqrt[5]*Pi^(3/4)), 10, 120][[1]] (* Vaclav Kotesovec, May 13 2023 *)

Equals exp(5*Pi/8) * Gamma(1/4) * (9 + 4*sqrt(5))^(1/4) * (-exp(5*Pi/2) + sqrt(exp(5*Pi) + 64*r^24))^(1/4) / (2^(7/4) * sqrt(5) * Pi^(3/4) * r^5), where r = A292904 = 1.00000015070175025002398949386987146797376100643050740569...

Equals exp(5*Pi/24) * Gamma(1/4) * (7 + 3*sqrt(5) + 12*sqrt(14*sqrt(5) - 30))^(1/8) / (2*sqrt(5)*Pi^(3/4)). - Vaclav Kotesovec, May 13 2023

A363018 Decimal expansion of Product_{k>=1} (1 - exp(-6Pik)).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, May 13 2023

			0.999999993487587821508587441627061243108330508136097236620870239066239...

Cf. A259148 phi(exp(-Pi)), A259149 phi(exp(-2*Pi)), A292888 phi(exp(-3*Pi)), A259150 phi(exp(-4*Pi)), A292905 phi(exp(-5*Pi)), A363117 phi(exp(-7*Pi)), A259151 phi(exp(-8*Pi)), A363118 phi(exp(-9*Pi)), A363019 phi(exp(-10*Pi)), A363081 phi(exp(-11*Pi)), A363020 phi(exp(-12*Pi)), A363178 phi(exp(-13*Pi)), A363119 phi(exp(-14*Pi)), A363179 phi(exp(-15*Pi)), A292864 phi(exp(-16*Pi)), A363120 phi(exp(-18*Pi)), A363021 phi(exp(-20*Pi)).

RealDigits[E^(Pi/4)*Gamma[1/4]*(2 - Sqrt[3])^(1/12)/(2*3^(3/8)*Pi^(3/4)), 10, 120][[1]]
RealDigits[QPochhammer[E^(-6*Pi)], 10, 120][[1]]

Equals exp(Pi/4) * Gamma(1/4) * (2 - sqrt(3))^(1/12) / (2 * 3^(3/8) * Pi^(3/4)).

Equals A292888 * A292887.

A363019 Decimal expansion of Product_{k>=1} (1 - exp(-10Pik)).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, May 13 2023

			0.999999999999977288989316758545823200993325029482707067413205453362995...

Cf. A259148 phi(exp(-Pi)), A259149 phi(exp(-2*Pi)), A292888 phi(exp(-3*Pi)), A259150 phi(exp(-4*Pi)), A292905 phi(exp(-5*Pi)), A363018 phi(exp(-6*Pi)), A363117 phi(exp(-7*Pi)), A259151 phi(exp(-8*Pi)), A363118 phi(exp(-9*Pi)), A363081 phi(exp(-11*Pi)), A363020 phi(exp(-12*Pi)), A363178 phi(exp(-13*Pi)), A363119 phi(exp(-14*Pi)), A363179 phi(exp(-15*Pi)), A292864 phi(exp(-16*Pi)), A363120 phi(exp(-18*Pi)), A363021 phi(exp(-20*Pi)).

RealDigits[E^(5*Pi/12)*Gamma[5/4]*Sqrt[2*(Sqrt[5] - 1)/5]/Pi^(3/4), 10, 120][[1]]
RealDigits[QPochhammer[E^(-10*Pi)], 10, 120][[1]]

Equals exp(5*Pi/12) * Gamma(5/4) * sqrt(2*(sqrt(5) - 1)/5) / Pi^(3/4).

Equals A292905 * A292904.

A363020 Decimal expansion of Product_{k>=1} (1 - exp(-12Pik)).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, May 13 2023

			0.999999999999999957588488169839222761089020220559669362727608370525037...

Cf. A259148 phi(exp(-Pi)), A259149 phi(exp(-2*Pi)), A292888 phi(exp(-3*Pi)), A259150 phi(exp(-4*Pi)), A292905 phi(exp(-5*Pi)), A363018 phi(exp(-6*Pi)), A363117 phi(exp(-7*Pi)), A259151 phi(exp(-8*Pi)), A363118 phi(exp(-9*Pi)), A363019 phi(exp(-10*Pi)), A363081 phi(exp(-11*Pi)), A363178 phi(exp(-13*Pi)), A363119 phi(exp(-14*Pi)), A363179 phi(exp(-15*Pi)), A292864 phi(exp(-16*Pi)), A363120 phi(exp(-18*Pi)), A363021 phi(exp(-20*Pi)).

RealDigits[E^(Pi/2) * Gamma[1/4] * (7 + 28*Sqrt[3] - 2*Sqrt[6*(-684 + 469*Sqrt[3])])^(1/24) / (2^(11/8)*3^(3/8)*Pi^(3/4)), 10, 120][[1]]
RealDigits[QPochhammer[E^(-12*Pi)], 10, 120][[1]]

Equals exp(Pi/2) * Gamma(1/4) * (7 + 28*sqrt(3) - 2*sqrt(6*(469*sqrt(3) - 684)))^(1/24) / (2^(11/8) * 3^(3/8) * Pi^(3/4)).

A363021 Decimal expansion of Product_{k>=1} (1 - exp(-20Pik)).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, May 13 2023

			0.999999999999999999999999999484209993745715964958151977112711625102369...

Cf. A259148 phi(exp(-Pi)), A259149 phi(exp(-2*Pi)), A292888 phi(exp(-3*Pi)), A259150 phi(exp(-4*Pi)), A292905 phi(exp(-5*Pi)), A363018 phi(exp(-6*Pi)), A363117 phi(exp(-7*Pi)), A259151 phi(exp(-8*Pi)), A363118 phi(exp(-9*Pi)), A363019 phi(exp(-10*Pi)), A363081 phi(exp(-11*Pi)), A363020 phi(exp(-12*Pi)), A363178 phi(exp(-13*Pi)), A363119 phi(exp(-14*Pi)), A363179 phi(exp(-15*Pi)), A292864 phi(exp(-16*Pi)), A363120 phi(exp(-18*Pi)).

RealDigits[E^(5*Pi/6) * Gamma[1/4] * (5^(1/4) - 1) * Sqrt[(Sqrt[5] - 1)/5] / (2^(19/8)*Pi^(3/4)), 10, 120][[1]]
RealDigits[QPochhammer[E^(-20*Pi)], 10, 120][[1]]

Equals exp(5*Pi/6) * Gamma(1/4) * (5^(1/4) - 1) * sqrt((sqrt(5) - 1)/5) / (2^(19/8) * Pi^(3/4)).

A363117 Decimal expansion of Product_{k>=1} (1 - exp(-7Pik)).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, May 15 2023

			0.999999999718573154172243658382901236462919560257076490298122086100117...

Cf. A259148 phi(exp(-Pi)), A259149 phi(exp(-2*Pi)), A292888 phi(exp(-3*Pi)), A259150 phi(exp(-4*Pi)), A292905 phi(exp(-5*Pi)), A363018 phi(exp(-6*Pi)), A259151 phi(exp(-8*Pi)), A363118 phi(exp(-9*Pi)), A363019 phi(exp(-10*Pi)), A363081 phi(exp(-11*Pi)), A363020 phi(exp(-12*Pi)), A363178 phi(exp(-13*Pi)), A363119 phi(exp(-14*Pi)), A363179 phi(exp(-15*Pi)), A292864 phi(exp(-16*Pi)), A363120 phi(exp(-18*Pi)), A363021 phi(exp(-20*Pi)).

RealDigits[E^(7*Pi/24) * Gamma[1/4] * ((Sqrt[5 - Sqrt[7]] - Sqrt[3*Sqrt[7] - 7]) * (2^(1/4) * Sqrt[5 + Sqrt[7]] + (56 + 23*Sqrt[7])^(1/4)))^(1/4) / (2^(19/16) * 7^(7/16) * Pi^(3/4)), 10, 120][[1]]
RealDigits[QPochhammer[E^(-7*Pi)], 10, 120][[1]]

Equals exp(7*Pi/24) * Gamma(1/4) * ((sqrt(5 - sqrt(7)) - sqrt(3*sqrt(7) - 7)) * (2^(1/4) * sqrt(5 + sqrt(7)) + (56 + 23*sqrt(7))^(1/4)))^(1/4) / (2^(19/16) * 7^(7/16) * Pi^(3/4)).

cp's OEIS Frontend

A259149 Decimal expansion of phi(exp(-2*Pi)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A259148 Decimal expansion of phi(exp(-Pi)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A259150 Decimal expansion of phi(exp(-4*Pi)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A259151 Decimal expansion of phi(exp(-8*Pi)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A292905 Decimal expansion of Product_{k>=1} (1 - exp(-5*Pi*k)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A363018 Decimal expansion of Product_{k>=1} (1 - exp(-6*Pi*k)).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A363019 Decimal expansion of Product_{k>=1} (1 - exp(-10*Pi*k)).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A363020 Decimal expansion of Product_{k>=1} (1 - exp(-12*Pi*k)).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A363021 Decimal expansion of Product_{k>=1} (1 - exp(-20*Pi*k)).

Views

A292905 Decimal expansion of Product_{k>=1} (1 - exp(-5Pik)).

A363018 Decimal expansion of Product_{k>=1} (1 - exp(-6Pik)).

A363019 Decimal expansion of Product_{k>=1} (1 - exp(-10Pik)).

A363020 Decimal expansion of Product_{k>=1} (1 - exp(-12Pik)).

A363021 Decimal expansion of Product_{k>=1} (1 - exp(-20Pik)).

A363117 Decimal expansion of Product_{k>=1} (1 - exp(-7Pik)).