A100220 -id:A100220 - cp's OEIS frontend

A048651 Decimal expansion of Product_{k >= 1} (1 - 1/2^k).

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

Offset: 0

N. J. A. Sloane

This is the limiting probability that a large random binary matrix is nonsingular (cf. A002884).
This constant is very close to 2^(13/24) * sqrt(Pi/log(2)) / exp(Pi^2/(6*log(2))) = 0.288788095086602421278899775042039398383022429351580356839... - Vaclav Kotesovec, Aug 21 2018
This constant is irrational (see Penn link). - Paolo Xausa, Dec 09 2024

			(1/2)*(3/4)*(7/8)*(15/16)*... = 0.288788095086602421278899721929230780088911904840685784114741...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 318, 354-361.

Harry J. Smith, Table of n, a(n) for n = 0..20000
Steven R. Finch, Digital Search Tree Constants. [Broken link]
Steven R. Finch, Digital Search Tree Constants. [From the Wayback machine]
Marvin Geiselhart, Ahmed Elkelesh, Moustafa Ebada, Sebastian Cammerer, and Stephan ten Brink, On the Automorphism Group of Polar Codes, arXiv:2101.09679 [cs.IT], 2021.
T. S. Jayram, Hellinger strikes back: a note on the multi-party information complexity of AND, LNCS 5687 (2009) 562-573.
Richard J. McIntosh, Some Asymptotic Formulae for q-Hypergeometric Series, Journal of the London Mathematical Society, Vol. 51, No. 1 (1995), pp. 120-136; alternative link.
Victor S. Miller, Counting Matrices that are Squares, arXiv:1606.09299 [math.GR], 2016.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
Michael Penn, A non-standard irrationality proof, YouTube video, 2024.
V. Arvind Rameshwar, Shreyas Jain, and Navin Kashyap, Sampling-Based Estimates of the Sizes of Constrained Subcodes of Reed-Muller Codes, arXiv:2309.08907 [cs.IT], 2023.
László Tóth, Alternating sums concerning multiplicative arithmetic functions, arXiv preprint arXiv:1608.00795 [math.NT], 2016.
Eric Weisstein's World of Mathematics, Infinite Product.
Eric Weisstein's World of Mathematics, Tree Searching.
Wikipedia, Pentagonal number theorem.
Wanchen Zhang, Yu Ning, Fei Shi, and Xiande Zhang, Extremal Maximal Entanglement, arXiv:2411.12208 [quant-ph], 2024. See p. 15.

Cf. A002884, A001318, A005327, A005329, A048652, A065446, A079555, A098844, A067080, A100220, A132019, A132020, A132026, A132038, A070933, A261584, A335764.

RealDigits[ Product[1 - 1/2^i, {i, 100}], 10, 111][[1]] (* Robert G. Wilson v, May 25 2011 *)
RealDigits[QPochhammer[1/2], 10, 100][[1]] (* Jean-François Alcover, Nov 18 2015 *)

default(realprecision, 20080); x=prodinf(k=1, -1/2^k, 1); x*=10; for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b048651.txt", n, " ", d)); \\ Harry J. Smith, May 07 2009

exp(-Sum_{k>0} sigma_1(k)/k*2^(-k)) = exp(-Sum_{k>0} A000203(k)/k*2^(-k)). - Hieronymus Fischer, Jul 28 2007
From Hieronymus Fischer, Aug 13 2007: (Start)
Equals lim inf Product_{k=0..floor(log_2(n))} floor(n/2^k)*2^k/n for n->oo.
Equals lim inf A098844(n)/n^(1+floor(log_2(n)))*2^(1/2*(1+floor(log_2(n)))*floor(log_2(n))) for n->oo.
Equals lim inf A098844(n)/n^(1+floor(log_2(n)))*2^A000217(floor(log_2(n))) for n->oo.
Equals lim inf A098844(n)/(n+1)^((1+log_2(n+1))/2) for n->oo.
Equals (1/2)*exp(-Sum_{n>0} 2^(-n)*Sum_{k|n} 1/(k*2^k)). (End)
Limit of A177510(n)/A000079(n-1) as n->infinity (conjecture). - Mats Granvik, Mar 27 2011
Product_{k >= 1} (1-1/2^k) = (1/2; 1/2){infinity}, where (a;q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Nov 27 2015
exp(Sum_{n>=1}(1/n/(1 - 2^n))) (according to Mathematica). - Mats Granvik, Sep 07 2016
(Sum_{k>0} (4^k-1)/(Product_{i=1..k} ((4^i-1)*(2*4^i-1))))*2 = 2/7 + 2/(3*7*31) + 2/(3*7*15*31*127)+2/(3*7*15*31*63*127*511) + ... (conjecture). - Werner Schulte, Dec 22 2016
Equals Sum_{k=-oo..oo} (-1)^k/2^((3*k+1)*k/2) (by Euler's pentagonal number theorem). - Amiram Eldar, Aug 13 2020
From Peter Bala, Dec 15 2020: (Start)
Constant C = Sum_{n >= 0} (-1)^n/( Product_{k = 1..n} (2^k - 1) ). The above conjectural result by Schulte follows by adding terms of this series in pairs.
C = (1/2)*Sum_{n >= 0} (-1/2)^n/( Product_{k = 1..n} (2^k - 1) ).
C = (3/8)*Sum_{n >= 0} (-1/4)^n/( Product_{k = 1..n} (2^k - 1) ).
1/C = Sum_{n >= 0} 2^(n*(n-1)/2)/( Product_{k = 1..n} (2^k - 1) ).
C = 1 - Sum_{n >= 0} (1/2)^(n+1)*Product_{k = 1..n} (1 - 1/2^k).
This latter identity generalizes as:
C = Sum_{n >= 0} (1/4)^(n+1)*Product_{k = 1..n} (1 - 1/2^k),
3*C = 1 - Sum_{n >= 0} (1/8)^(n+1)*Product_{k = 1..n} (1 - 1/2^k),
3*7*C = 6 + Sum_{n >= 0} (1/16)^(n+1)*Product_{k = 1..n} (1 - 1/2^k),
3*7*15*C = 91 - Sum_{n >= 0} (1/32)^(n+1)*Product_{k = 1..n} (1 - 1/2^k),
and so on, where the sequence [1, 0, 1, 6, 91, ...] is A005327.
(End)
From Amiram Eldar, Feb 19 2022: (Start)
Equals sqrt(2*Pi/log(2)) * exp(log(2)/24 - Pi^2/(6*log(2))) * Product_{k>=1} (1 - exp(-4*k*Pi^2/log(2))) (McIntosh, 1995).
Equals Sum_{n>=0} (-1)^n/A005329(n).
Equals exp(-A335764). (End)
Equals 1/A065446. - Hugo Pfoertner, Nov 23 2024

Corrected by Hieronymus Fischer, Jul 28 2007

A132038 Decimal expansion of Product_{k>0} (1-1/10^k).

Original entry on oeis.org

8, 9, 0, 0, 1, 0, 0, 9, 9, 9, 9, 8, 9, 9, 9, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 9, 9, 9, 9, 9, 9, 9, 9, 8, 9, 9, 9, 9, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 8, 9, 9, 9, 9, 9, 9, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 9, 9, 9

Offset: 0

Hieronymus Fischer, Aug 14 2007

			0.8900100999989990000001000...

G. C. Greubel, Table of n, a(n) for n = 0..1500
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge University Press, 2009, page 49.
Richard J. McIntosh, Some Asymptotic Formulae for q-Hypergeometric Series, Journal of the London Mathematical Society, Vol. 51, No. 1 (1995), pp. 120-136; alternative link.

Cf. A000203, A027878, A048651, A067080, A098844, A100220, A132019, A132026.

digits = 105; Clear[p]; p[n_] := p[n] = RealDigits[Product[1-1/10^k , {k, 1, n}], 10, digits] // First; p[10]; p[n=20]; While[p[n] != p[n/2], n = 2*n]; p[n] (* Jean-François Alcover, Feb 17 2014 *)
RealDigits[QPochhammer[1/10], 10, 105][[1]] (* Jean-François Alcover, Nov 18 2015 *)
N[QPochhammer[1/10,1/10]] (* G. C. Greubel, Nov 30 2015 *)

prodinf(x=1,-.1^x,1) \\ Charles R Greathouse IV, Nov 16 2013

Equals exp( -Sum_{n>0} sigma_1(n)/(n*10^n) ).
Equals (1/10; 1/10){infinity}, where (a; q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Nov 30 2015
From Amiram Eldar, May 09 2023: (Start)
Equals sqrt(2*Pi/log(10)) * exp(log(10)/24 - Pi^2/(6*log(10))) * Product_{k>=1} (1 - exp(-4*k*Pi^2/log(10))) (McIntosh, 1995).
Equals Sum_{n>=0} (-1)^n/A027878(n). (End)

A100221 Decimal expansion of Product_{k>=1} (1-1/4^k).

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Nov 09 2004

			0.68853753712033971545651435729350818467554981937833...

G. C. Greubel, Table of n, a(n) for n = 0..1200
Richard J. McIntosh, Some Asymptotic Formulae for q-Hypergeometric Series, Journal of the London Mathematical Society, Vol. 51, No. 1 (1995), pp. 120-136; alternative link.
Eric Weisstein's World of Mathematics, Infinite Product.
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol.

Cf. A027637, A048651, A100220, A100222.

Cf. A000203, A132020, A132034, A132035, A132036, A132037, A132038, A258459.

EllipticThetaPrime[1, 0, 1/2]^(1/3)/2^(1/4)
N[QPochhammer[1/4]] (* G. C. Greubel, Nov 30 2015 *)
RealDigits[Fold[Times,1-1/4^Range[1000]],10,110][[1]] (* Harvey P. Dale, Sep 27 2024 *)

prodinf(x=1, 1-1/4^x) \\ Altug Alkan, Dec 01 2015

Equals exp(-Sum_{k>0} sigma_1(k)/(k*4^k)) where sigma_1() is A000203(). - Hieronymus Fischer, Aug 07 2007
Equals (1/4; 1/4){infinity}, where (a;q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Nov 30 2015
From Amiram Eldar, May 09 2023: (Start)
Equals sqrt(Pi/log(2)) * exp(log(2)/12 - Pi^2/(12*log(2))) * Product_{k>=1} (1 - exp(-2*k*Pi^2/log(2))) (McIntosh, 1995).
Equals Sum_{n>=0} (-1)^n/A027637(n). (End)

A053290 Number of nonsingular n X n matrices over GF(3).

Original entry on oeis.org

1, 2, 48, 11232, 24261120, 475566474240, 84129611558952960, 134068444202678083338240, 1923442429811445711790394572800, 248381049201184165590947520186915225600, 288678833735376059528974260112416365258106470400

Offset: 0

Stephen G Penrice, Mar 04 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..45
Jeffrey Overbey, William Traves, and Jerzy Wojdylo, On the Keyspace of the Hill Cipher, Cryptologia, Vol. 29, Iss. 1 (2005), pp. 59-72; author's copy.

Column k=3 of A316622.
Cf. A002884, A003787, A027871, A047656, A053291, A053292, A053293.
Cf. A100220, A219205.

[1] cat [&*[(3^n - 3^k): k in [0..n-1]]: n in [1..9]]; // Bruno Berselli, Jan 28 2013

Table[Product[3^n - 3^k, {k, 0, n - 1}], {n, 0, 10}] (* Geoffrey Critzer, Jan 26 2013; edited by Vincenzo Librandi, Jan 28 2013 *)

for(n=0,10, print1(prod(k=0,n-1, 3^n - 3^k), ", ")) \\ G. C. Greubel, May 31 2018

a(n) = Product_{k=0..n-1}(3^n-3^k). - corrected by Michel Marcus, Sep 18 2015
a(n) = A047656(n)*A027871(n). - Bruno Berselli, Jan 30 2013
From Amiram Eldar, Jul 06 2025: (Start)
a(n) = Product_{k=1..n} A219205(k).
a(n) ~ c * 3^(n^2), where c = A100220. (End)

More terms from Vladeta Jovovic, Mar 16 2000

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

Original entry on oeis.org

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

A132027 a(n) = Product_{k=0..floor(log_3(n))} floor(n/3^k), n>=1.

Original entry on oeis.org

1, 2, 3, 4, 5, 12, 14, 16, 27, 30, 33, 48, 52, 56, 75, 80, 85, 216, 228, 240, 294, 308, 322, 384, 400, 416, 729, 756, 783, 900, 930, 960, 1089, 1122, 1155, 1728, 1776, 1824, 2028, 2080, 2132, 2352, 2408, 2464, 3375, 3450, 3525, 3840, 3920, 4000, 4335

Offset: 1

Hieronymus Fischer, Aug 13 2007, Aug 20 2007

nonn

If n is written in base 3 as n=d(m)d(m-1)d(m-2)...d(2)d(1)d(0) (where d(k) is the digit at position k) then a(n) is also the product d(m)d(m-1)d(m-2)...d(2)d(1)d(0)*d(m)d(m-1)d(m-2)...d(2)d(1)*d(m)d(m-1)d(m= -2)...d(2)*...*d(m)d(m-1)d(m-2)*d(m)d(m-1)*d(m).

			a(11)=floor(11/3^0)*floor(11/3^1)*floor(11/3^2)=11*3*1=33;
a(13)=52 since 13=111(base-3) and so a(13)=111*11*1(base-3)=13*4*1=52.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A048651, A098844, A067080, A132019, A100220, A000217.
For formulas regarding a general parameter p (i.e. terms floor(n/p^k)) see A132264.
For the product of terms floor(n/p^k) for p=2 to p=12 see A098844(p=2), A132028(p=4)-A132033(p=9), A067080(p=10), A132263(p=11), A132264(p=12).
For the products of terms 1+floor(n/p^k) see A132269-A132272, A132327, A132328.

Table[(f = If[# < 3, #, #*f[Quotient[#, 3]]] &)[n], {n, 51}] (* Ivan Neretin, May 29 2016 *)

Recurrence: a(n)=n*a(floor(n/3)); a(n*3^m)=n^m*3^(m(m+1)/2)*a(n).
a(k*3^m)=k^(m+1)*3^(m(m+1)/2), for k=1 or 2.
a(n)<=b(n), where b(n)=n^(1+floor(log_3(n)))/3^(1/2*(1+floor(log_3(n)))*floor(log_3(n))); equality holds if n is a power of 3 or two times a power of 3.
Also: a(n)<=2^((1-log_3(2))/2)*n^((1+log_3(n))/2)=1.1364507...*3^A000217(log_3(n)), equality for n=2*3^m, m>=0.
a(n)>c*b(n), where c=0.3826631966790330232889550... (see constant A132019).
Also: a(n)>c*2^((1-log_3(2))/2)*n^((1+log_3(n))/2)=0.434877...*3^A000217(log_3(n)).
lim inf a(n)/b(n)=0.3826631966790330232889550..., for n-->oo.
lim sup a(n)/b(n)=1, for n-->oo.
lim inf a(n)/n^((1+log_3(n))/2)=0.3826631966790330232889550...*sqrt(2)/2^log_3(sqrt(2)), for n-->oo.
lim sup a(n)/n^((1+log_3(n))/2)=sqrt(2)/2^log_3(sqrt(2)), for n-->oo.
lim inf a(n)/a(n+1)=0.3826631966790330232889550... for n-->oo (see constant A132019).
a(n)=O(n^((1+log_3(n))/2)).

A027871 a(n) = Product_{i=1..n} (3^i - 1).

Original entry on oeis.org

1, 2, 16, 416, 33280, 8053760, 5863137280, 12816818094080, 84078326697164800, 1654829626053597593600, 97714379759212830706892800, 17309711516825516108403231948800

Offset: 0

N. J. A. Sloane

nonn

2*(10)^m|a(n) where 4*m <= n <= 4*m+3 for m >= 1. - G. C. Greubel, Nov 20 2015

Given probability p = 1/3^n that an outcome will occur at the n-th stage of an infinite process, then starting at n=1, 1-a(n)/A047656(n+1) is the probability that the outcome has occurred at or before the n-th iteration. The limiting ratio is 1-A100220 ~ 0.4398739. - Bob Selcoe, Mar 01 2016

G. C. Greubel, Table of n, a(n) for n = 0..50

Cf. A005329 (q=2), A027637 (q=4), A027872 (q=5), A027873 (q=6), A027875 (q=7), A027876 (q=8), A027877 (q=9), A027878 (q=10), A027879 (q=11), A027880 (q=12).

Cf. A047656, A100220, A132324.

[1] cat [&*[ 3^k-1: k in [1..n] ]: n in [1..11]]; // Vincenzo Librandi, Dec 24 2015

A027871 := proc(n)
    mul( 3^i-1,i=1..n) ;
end proc:
seq(A027871(n),n=0..8) ; # R. J. Mathar, Jul 13 2017

Table[Product[(3^k-1),{k,1,n}],{n,0,20}] (* Vaclav Kotesovec, Jul 17 2015 *)
Abs@QPochhammer[3, 3, Range[0, 10]] (* Vladimir Reshetnikov, Nov 20 2015 *)

a(n) = prod(i=1, n, 3^i-1); \\ Michel Marcus, Nov 21 2015

a(n) ~ c * 3^(n*(n+1)/2), where c = A100220 = Product_{k>=1} (1-1/3^k) = 0.560126077927948944969792243314140014379736333798... . - Vaclav Kotesovec, Nov 21 2015
a(n) = 3^(binomial(n+1,2))*(1/3;1/3){n}, where (a;q){n} is the q-Pochhammer symbol. - G. C. Greubel, Dec 24 2015
a(n) = Product_{i=1..n} A024023(i). - Michel Marcus, Dec 27 2015
G.f.: Sum_{n>=0} 3^(n*(n+1)/2)*x^n / Product_{k=0..n} (1 + 3^k*x). - Ilya Gutkovskiy, May 22 2017
From Amiram Eldar, Feb 19 2022: (Start)
Sum_{n>=0} 1/a(n) = A132324.
Sum_{n>=0} (-1)^n/a(n) = A100220. (End)

A132034 Decimal expansion of Product_{k>0} (1-1/6^k).

Original entry on oeis.org

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

Offset: 0

Hieronymus Fischer, Aug 14 2007

			0.805687728162164940923750...

G. C. Greubel, Table of n, a(n) for n = 0..1200
Richard J. McIntosh, Some Asymptotic Formulae for q-Hypergeometric Series, Journal of the London Mathematical Society, Vol. 51, No. 1 (1995), pp. 120-136; alternative link.

Cf. A000203, A027873, A048651, A067080, A098844, A100220, A132022, A132026, A132038.

digits = 103; NProduct[1-1/6^k, {k, 1, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+20] // N[#, digits+20]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
N[QPochhammer[1/6,1/6]] (* G. C. Greubel, Nov 30 2015 *)

prodinf(x=1, 1-(1/6)^x) \\ Altug Alkan, Dec 01 2015

Equals exp( -Sum_{n>0} sigma_1(n)/(n*6^n) ).
Equals (1/6; 1/6){infinity}, where (a;q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Nov 30 2015
From Amiram Eldar, May 09 2023: (Start)
Equals sqrt(2*Pi/log(6)) * exp(log(6)/24 - Pi^2/(6*log(6))) * Product_{k>=1} (1 - exp(-4*k*Pi^2/log(6))) (McIntosh, 1995).
Equals Sum_{n>=0} (-1)^n/A027873(n). (End)

cp's OEIS Frontend

A048651 Decimal expansion of Product_{k >= 1} (1 - 1/2^k).

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A132038 Decimal expansion of Product_{k>0} (1-1/10^k).

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A100221 Decimal expansion of Product_{k>=1} (1-1/4^k).

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A053290 Number of nonsingular n X n matrices over GF(3).

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

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

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

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A132027 a(n) = Product_{k=0..floor(log_3(n))} floor(n/3^k), n>=1.