A132019 -id:A132019 - cp's OEIS frontend

A132022 Decimal expansion of Product_{k>=0} (1 - 1/(2*6^k)).

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

Offset: 0

Hieronymus Fischer, Aug 14 2007

			0.45071262522603913...

G. C. Greubel, Table of n, a(n) for n = 0..2000

Cf. A000217, A048651, A098844, A067080, A132019, A132026, A132030, A132034, A167747.

digits = 103; NProduct[1-1/(2*6^k), {k, 0, Infinity}, NProductFactors -> 200, WorkingPrecision -> digits+5] // N[#, digits+5]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
(1/2)*N[QPochhammer[1/12, 1/6], 200] (* G. C. Greubel, Dec 20 2015 *)

prodinf(x=0, 1-1/(2*6^x)) \\ Altug Alkan, Dec 20 2015

Equals lim inf_{n->oo} Product_{k=0..floor(log_6(n))} floor(n/6^k)*6^k/n.
Equals lim inf_{n->oo} A132030(n)/n^(1+floor(log_6(n)))*6^(1/2*(1+floor(log_6(n)))*floor(log_6(n))).
Equals lim inf_{n->oo} A132030(n)/n^(1+floor(log_6(n)))*6^A000217(floor(log_6(n))).
Equals (1/2)*exp(-Sum_{n>0} 6^(-n)*Sum{k|n} 1/(k*2^k)).
Equals lim inf_{n->oo} A132030(n)/A132030(n+1).
Equals (1/2)*(1/12; 1/6){infinity}, where (a;q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Dec 20 2015
Equals Product_{n>=1} (1 - 1/A167747(n)). - Amiram Eldar, May 09 2023

A132024 Decimal expansion of Product_{k>=0} (1-1/(2*8^k)).

Original entry on oeis.org

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

Offset: 0

Hieronymus Fischer, Aug 14 2007

			0.46456888368647639098...

Cf. A048651, A098844, A067080, A132019, A132026, A132032, A132036, A000217, A013730.

RealDigits[QPochhammer[1/2,1/8],10,120][[1]] (* Harvey P. Dale, May 23 2011 *)

prodinf(k=0, 1 - 1/(2*8^k)) \\ Amiram Eldar, May 09 2023

Equals lim inf_{n->oo} Product_{k=0..floor(log_8(n))} floor(n/8^k)*8^k/n.
Equals lim inf_{n->oo} A132032(n)/n^(1+floor(log_8(n)))*8^(1/2*(1+floor(log_8(n)))*floor(log_8(n))).
Equals lim inf_{n->oo} A132032(n)/n^(1+floor(log_8(n)))*8^A000217(floor(log_8(n))).
Equals (1/2)*exp(-Sum_{n>0} 8^(-n)*Sum_{k|n} 1/(k*2^k)).
Equals lim inf_{n->oo} A132032(n)/A132032(n+1).
Equals Product_{n>=0} (1 - 1/A013730(n)). - Amiram Eldar, May 09 2023

Name corrected by Amiram Eldar, May 09 2023

A132023 Decimal expansion of Product_{k>=0} 1-1/(2*7^k).

Original entry on oeis.org

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

Offset: 0

Hieronymus Fischer, Aug 14 2007

			0.4587667266997689850200...

Cf. A048651, A098844, A067080, A132019, A132026, A132031, A132035, A000217, A109808.

digits = 103; NProduct[1-1/(2*7^k), {k, 0, Infinity}, NProductFactors -> 200, WorkingPrecision -> digits+5] // N[#, digits+5]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
RealDigits[QPochhammer[1/2, 1/7], 10, 120][[1]] (* Amiram Eldar, May 08 2023 *)

Equals lim inf_{n->oo} Product_{k=0..floor(log_7(n))} floor(n/7^k)*7^k/n.
Equals lim inf_{n->oo} A132031(n)/n^(1+floor(log_7(n)))*7^(1/2*(1+floor(log_7(n)))*floor(log_7(n))).
Equals lim inf_{n->oo} A132031(n)/n^(1+floor(log_7(n)))*7^A000217(floor(log_7(n))).
Equals 1/2*exp(-Sum_{n>0} 7^(-n)*Sum_{k|n} 1/(k*2^k)).
Equals lim inf_{n->oo} A132031(n)/A132031(n+1).
Equals Product_{n>=1} (1 - 1/A109808(n)). - Amiram Eldar, May 08 2023

A132025 Decimal expansion of Product_{k>=0} 1-1/(2*9^k).

Original entry on oeis.org

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

Offset: 0

Hieronymus Fischer, Aug 14 2007

			0.4689451783670236932832800...

Cf. A048651, A098844, A067080, A132019, A132026, A132033, A132037, A000217, A270369.

digits = 103; NProduct[1-1/(2*9^k), {k, 0, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+5] // N[#, digits+5]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
RealDigits[QPochhammer[1/2, 1/9], 10, 120][[1]] (* Amiram Eldar, May 08 2023 *)

Equals lim inf_{n->oo} Product_{k=0..floor(log_9(n))} floor(n/9^k)*9^k/n.
Equals lim inf_{n->oo} A132033(n)/n^(1+floor(log_9(n)))*9^(1/2*(1+floor(log_9(n)))*floor(log_9(n))).
Equals lim inf_{n->oo} A132033(n)/n^(1+floor(log_9(n)))*9^A000217(floor(log_9(n))).
Equals (1/2)*exp(-Sum_{n>0} 9^(-n)*Sum_{k|n} 1/(k*2^k)).
Equals lim inf_{n->oo} A132033(n)/A132033(n+1).
Equals Product_{n>=1} (1 - 1/A270369(n)). - Amiram Eldar, May 08 2023

A132021 Decimal expansion of Product_{k>=0} 1-1/(2*5^k).

Original entry on oeis.org

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

Offset: 0

Hieronymus Fischer, Aug 14 2007

			0.438796837203638531...

Cf. A048651, A098844, A067080, A132019, A132026, A132029, A100222, A000217, A020729.

digits = 103; NProduct[1-1/(2*5^k), {k, 0, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+5] // N[#, digits+5]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
RealDigits[QPochhammer[1/2, 1/5], 10, 120][[1]] (* Amiram Eldar, May 08 2023 *)

Equals lim inf_{n->oo} Product_{k=0..floor(log_5(n))} floor(n/5^k)*5^k/n.
Equals lim inf_{n->oo} A132029(n)/n^(1+floor(log_5(n)))*5^(1/2*(1+floor(log_5(n)))*floor(log_5(n))).
Equals lim inf_{n->oo} A132029(n)/n^(1+floor(log_5(n)))*5^A000217(floor(log_5(n))).
Equals (1/2)*exp(-Sum_{n>0} 5^(-n)*Sum_{k|n} 1/(k*2^k)).
Equals lim inf_{n->oo} A132029(n)/A132029(n+1).
Equals Product_{n>=0} (1 - 1/A020729(n)). - Amiram Eldar, May 08 2023

A132266 Decimal expansion of Product_{k>=0} (1 - 1/(2*12^k)).

Original entry on oeis.org

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

Offset: 0

Hieronymus Fischer, Aug 20 2007

			0.47735217025489380198334286365820...

G. C. Greubel, Table of n, a(n) for n = 0..2000

Cf. A000217, A048651, A132019-A132026, A132034-A132038, A132265-A132268, A132323-A132326, A132264.

digits = 105; NProduct[1-1/(2*12^k), {k, 0, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+5] // N[#, digits+5]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
(1/2)*N[QPochhammer[1/24, 1/12], 200] (* G. C. Greubel, Dec 20 2015 *)

prodinf(x=0, 1-1/(2*12^x)) \\ Altug Alkan, Dec 20 2015

lim inf (Product_{k=0..floor(log_12(n))} floor(n/12^k)*12^k/n) for n-->oo.
lim inf A132264(n)*12^((1+floor(log_12(n)))*floor(log_12(n))/2)/n^(1+floor(log_12(n))) for n-->oo.
lim inf A132264(n)*12^A000217(floor(log_12(n)))/n^(1+floor(log_12(n))) for n-->oo.
(1/2)*exp(-Sum_{n>0} 12^(-n)*Sum_{k|n} 1/(k*2^k)).
lim inf A132264(n)/A132264(n+1) = 0.47735217025489380... for n-->oo.
Equals (1/2)*(1/24; 1/12){infinity}, where (a;q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Dec 20 2015

cp's OEIS Frontend

A132022 Decimal expansion of Product_{k>=0} (1 - 1/(2*6^k)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A132024 Decimal expansion of Product_{k>=0} (1-1/(2*8^k)).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A132023 Decimal expansion of Product_{k>=0} 1-1/(2*7^k).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A132025 Decimal expansion of Product_{k>=0} 1-1/(2*9^k).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A132021 Decimal expansion of Product_{k>=0} 1-1/(2*5^k).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A132266 Decimal expansion of Product_{k>=0} (1 - 1/(2*12^k)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula