A100220 -id:A100220 - cp's OEIS frontend

A132035 Decimal expansion of Product_{k>0} (1-1/7^k).

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

Offset: 0

Hieronymus Fischer, Aug 14 2007

			0.8367954070890378710...

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. A027875, A048651, A100220, A132023, A132026, A132038, A098844, A067080, A000203.

digits = 103; NProduct[1-1/7^k, {k, 1, Infinity}, NProductFactors -> 200, WorkingPrecision -> digits+3] // N[#, digits+3]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
RealDigits[QPochhammer[1/7], 10, 100][[1]] (* Amiram Eldar, May 09 2023 *)

prodinf(k=1, 1 - 1/(7^k)) \\ Amiram Eldar, May 09 2023

Equals exp(-Sum_{n>0} sigma_1(n)/(n*7^n)) = exp(-Sum_{n>0} A000203(n)/(n*7^n)).
From Amiram Eldar, May 09 2023: (Start)
Equals sqrt(2*Pi/log(7)) * exp(log(7)/24 - Pi^2/(6*log(7))) * Product_{k>=1} (1 - exp(-4*k*Pi^2/log(7))) (McIntosh, 1995).
Equals Sum_{n>=0} (-1)^n/A027875(n). (End)

A015001 q-factorial numbers for q=3.

Original entry on oeis.org

1, 1, 4, 52, 2080, 251680, 91611520, 100131391360, 328430963660800, 3232089113385932800, 95424198983606279987200, 8452007576574959037306265600, 2245867453247498115393020895232000, 1790317944898228845164815929864036352000

Offset: 0

Olivier Gérard

a(n) is the number of maximal chains in the lattice of subspaces of an n-dimensional vector space over GF(3). - Geoffrey Critzer, Sep 07 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..50
Index entries for sequences related to factorial numbers.

Cf. A015002, A015004, A015005, A015006, A015007, A015008, A015009, A015011.
Column q=3 of A069777.
Cf. A000079, A003462, A047656, A053290, A100220.

[n le 1 select 1 else (3^n-1)*Self(n-1)/2: n in [1..15]]; // Vincenzo Librandi, Oct 22 2012

RecurrenceTable[{a[1]==1, a[n]==((3^n - 1) * a[n-1])/2}, a, {n,15}] (* Vincenzo Librandi, Oct 27 2012 *)
Table[QFactorial[n, 3], {n, 15}] (* Bruno Berselli, Aug 14 2013 *)

a(n) = Product_{k=1..n} (q^k - 1) / (q - 1).
a(0) = 1, a(n) = (3^n - 1)*a(n-1)/2. - Vincenzo Librandi, Oct 27 2012
a(n) = (Product_{i=0..n-1} (q^n-q^i))/((q-1)^n*q^binomial(n,2)) = A053290(n)/(A000079(n)*A047656(n)). - Geoffrey Critzer, Sep 07 2022
From Amiram Eldar, Jul 05 2025: (Start)
a(n) = Product_{k=1..n} A003462(k).
a(n) ~ c * 3^(n*(n+1)/2)/2^n, where c = A100220. (End)

a(0)=1 prepended by Alois P. Heinz, Sep 08 2021

A259147 Decimal expansion of phi(exp(-Pi/2)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 19 2015

			0.74931147780000278742962565878338031190409252790117392831206731...

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)), 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)), A363019 phi(exp(-10*Pi)), A363020 phi(exp(-12*Pi)), A292864 phi(exp(-16*Pi)), A363021 phi(exp(-20*Pi)).

Cf. A292819, A292823, A292827.

phi[q_] := QPochhammer[q, q]; RealDigits[phi[Exp[-Pi/2]], 10, 105] // 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/2)) = ((sqrt(2) - 1)^(1/3)*(4 + 3*sqrt(2))^(1/24) * exp(Pi/48) * Gamma(1/4))/(2^(5/6)*Pi^(3/4)).
phi(exp(-Pi/2)) = (sqrt(2)-1)^(1/4) * exp(Pi/48) * Gamma(1/4)/(2^(13/16)*Pi^(3/4)). - Vaclav Kotesovec, Jul 03 2017

A003793 Order of (usually) simple Chevalley group A_n (3).

Original entry on oeis.org

1, 12, 5616, 6065280, 237783237120, 21032402889738240, 67034222101339041669120, 480860607452861427947598643200, 124190524600592082795473760093457612800

Offset: 0

N. J. A. Sloane

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites], p. xvi.
H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 131.

Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Robert Steinberg, Lectures on Chevalley Groups, Dept. of Mathematics, Yale University, 1967, p. 130-131.
Index entries for sequences related to groups.

Cf. A003787, A003794, A003795, A003797, A003798, A003799, A100220.

f[m_, n_] := Block[{g, x, y}, g[x_, y_] := x^(y (y + 1)/2) Product[x^k - 1, {k, 2, y + 1}]; g[m, n]/GCD[n + 1, m - 1]]; f[3, #] & /@ Range[0, 8] (* Michael De Vlieger, Sep 18 2015 *)

a(n) = a(3,n) where a(q,n) = A(q,n) / gcd(n+1, q-1) and A(q,n) is defined in A003787. - Sean A. Irvine, Sep 18 2015

a(n) ~ c * gcd(n, 2) * 3^(n^2 + 2*n), where c = (3/4) * A100220. - Amiram Eldar, Jul 11 2025

A132268 Decimal expansion of Product_{k>0} (1-1/12^k).

Original entry on oeis.org

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

Offset: 0

Hieronymus Fischer, Aug 20 2007

			0.9097262689059948886363620469770...

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. A027880, A048651, A100220, A132019-A132026, A132034-A132038, A132265-A132267, A132323-A132326, A000203.

digits = 106; NProduct[1-1/12^k, {k, 1, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+3] // N[#, digits+3]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
N[QPochhammer[1/12,1/12]] (* G. C. Greubel, Dec 05 2015 *)

prodinf(k=1, (1-1/12^k)) \\ Michel Marcus, Dec 05 2015

Equals exp(-Sum_{n>0} sigma_1(n)/(n*12^n)) = exp(-Sum_{n>0} A000203(n)/(n*12^n)).
Equals (1/12; 1/12){infinity}, where (a;q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Dec 05 2015
From Amiram Eldar, May 09 2023: (Start)
Equals sqrt(2*Pi/log(12)) * exp(log(12)/24 - Pi^2/(6*log(12))) * Product_{k>=1} (1 - exp(-4*k*Pi^2/log(12))) (McIntosh, 1995).
Equals Sum_{n>=0} (-1)^n/A027880(n). (End)

A028665 Galois numbers for p=3; order of group AGL(n,3).

Original entry on oeis.org

1, 6, 432, 303264, 1965150720, 115562653240320, 61330486826476707840, 293207687471256968260730880, 12619705781992895315056778792140800, 4888884191426907931326620039839052385484800, 17046196453240220939126401085378073952125928970649600

Offset: 0

Olivier Gérard

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..45
Index entries for sequences related to groups.

Cf. A028365, A028667, A028669, A028673, A028675, A028679, A028681, A028685.

Cf. A100220.

FoldList[ #1*3^#2 (3^#2-1)&, 1, Range[ 20 ]]
a[n_] := 3^n * Product[3^n - 3^k, {k, 0, n-1}]; Array[a, 11, 0] (* Amiram Eldar, Jul 12 2025 *)

a(n) = 3^n * prod(k = 0, n-1, 3^n - 3^k); \\ Amiram Eldar, Jul 12 2025

a(n) = 3^n * Product_{k=0..n-1} (3^n - 3^k).

a(n) ~ c * 3^(n^2+n), where c = A100220. - Amiram Eldar, Jul 12 2025

A003790 Order of universal Chevalley group A_n (7).

Original entry on oeis.org

1, 336, 5630688, 4635182361600, 187035198320488089600, 369826556020831611935738265600, 35832085525362833262818017603275320524800, 170115000551935077294273059250893063598899496222720000

Offset: 0

N. J. A. Sloane

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites], p. xvi.
H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 131.

Index entries for sequences related to groups.

Cf. A003787, A003788, A003789, A003791, A003792,A053293, A100220.

[&*[(7^n - 7^k): k in [0..n-1]]/6: n in [1..10]]; // Vincenzo Librandi, Sep 19 2015

f[m_, n_] := m^(n (n + 1)/2) Product[m^k - 1, {k, 2, n + 1}]; f[7, #] & /@ Range[0, 7] (* Michael De Vlieger, Sep 18 2015 *)

Numbers so far appear to equal A053293(n)/6. - Ralf Stephan, Mar 30 2004
a(n) = A(7,n) where A(q,n) is defined in A003787. - Sean A. Irvine, Sep 18 2015
a(n) ~ c * 7^(n*(n+2)), where c = (7/6) * A100220 = 0.840189116891... . - Amiram Eldar, Jul 07 2025

a(7) from Sean A. Irvine, Sep 18 2015

A132267 Decimal expansion of Product_{k>0} (1-1/11^k).

Original entry on oeis.org

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

Offset: 0

Hieronymus Fischer, Aug 20 2007

			0.900832706809715279949862694760...

G. C. Greubel, Table of n, a(n) for n = 0..2000
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, A027879, A048651, A100220, A132019-A132026, A132034-A132038, A132265-A132268, A132323, A132324, A132325, A132326.

digits = 105; NProduct[1-1/11^k, {k, 1, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+3] // N[#, digits+3]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
N[QPochhammer[1/11, 1/11], 200] (* G. C. Greubel, Dec 20 2015 *)

prodinf(x=1, 1-1/11^x) \\ Altug Alkan, Dec 20 2015

Equals exp(-Sum_{n>0} sigma_1(n)/(n*11^n)) = exp(-Sum_{n>0} A000203(n)/(n*11^n)).
Equals (1/11; 1/11){infinity}, where (a;q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Dec 20 2015
From Amiram Eldar, May 09 2023: (Start)
Equals sqrt(2*Pi/log(11)) * exp(log(11)/24 - Pi^2/(6*log(11))) * Product_{k>=1} (1 - exp(-4*k*Pi^2/log(11))) (McIntosh, 1995).
Equals Sum_{n>=0} (-1)^n/A027879(n). (End)

A007951 Ternary sieve: delete every 3rd number, then every 9th, 27th, etc.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 10, 11, 14, 16, 17, 19, 20, 22, 23, 25, 28, 29, 31, 32, 34, 35, 37, 38, 41, 43, 46, 47, 49, 50, 52, 55, 56, 58, 59, 61, 62, 64, 65, 68, 70, 71, 73, 74, 76, 77, 79, 82, 83, 85, 86, 88, 91, 92, 95, 97, 98, 100, 101, 103, 104, 106, 109, 110, 112, 113, 115, 116, 118

Offset: 1

R. Muller

nonn

Smarandache calls this a "trinary" sieve. - N. J. A. Sloane, Jan 03 2020

The asymptotic density of this sequence is Product_{k>=1} (1 - 1/3^k) = 0.560126... (A100220). - Amiram Eldar, Mar 21 2021

Rémy Sigrist, Table of n, a(n) for n = 1..25000
Florentin Smarandache, Only Problems, Not Solutions!, 4th ed., 1993; Problem 96.
Index entries for sequences generated by sieves

Cf. A007950, A092418, A100220.

v = List([1..118]); t=3; while (#v>=t, forstep (k=#v\t, 1, -1, listpop(v, k*t);); t*=3;); print (v) \\ Rémy Sigrist, Jan 05 2020

A132324 Decimal expansion of Product_{k>=1} (1+1/3^k).

Original entry on oeis.org

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

Offset: 1

Hieronymus Fischer, Aug 20 2007

Half the constant A132323.

			1.56493401856701153793884910...

G. C. Greubel, Table of n, a(n) for n = 1..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. A027871, A079555, A100220, A132019-A132026, A132034-A132038, A132265-A132268, A132323-A132326, A132327, A132328, A000593.

digits = 105; NProduct[1+1/3^k, {k, 1, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+5] // N[#, digits+5]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
N[QPochhammer[-1/3,1/3]] (* G. C. Greubel, Dec 01 2015 *)

(1/2)*lim sup Product{k=0..floor(log_3(n))} (1+1/floor(n/3^k)) for n-->oo.
(1/2)*lim sup A132327(n)/A132027(n) for n-->oo.
(1/2)*lim sup A132327(n)/n^((1+log_3(n))/2) for n-->oo.
(1/2)*lim sup A132328(n)/n^((log_3(n)-1)/2) for n-->oo.
exp(Sum_{n>0} 3^(-n)*Sum_{k|n} -(-1)^k/k) = exp(Sum_{n>0} A000593(n)/(n*3^n)).
(1/2)*lim sup A132327(n+1)/A132327(n) = 1.56493401856701153793884910... for n-->oo.
Equals (-1/3; 1/3){infinity}, where (a;q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Dec 01 2015
From Amiram Eldar, Feb 19 2022: (Start)
Equals (sqrt(2)/2) * exp(log(3)/24 + Pi^2/(12*log(3))) * Product_{k>=1} (1 - exp(-2*(2*k-1)*Pi^2/log(3))) (McIntosh, 1995).
Equals Sum_{n>=0} 1/A027871(n). (End)

cp's OEIS Frontend

A132035 Decimal expansion of Product_{k>0} (1-1/7^k).

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A015001 q-factorial numbers for q=3.

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A259147 Decimal expansion of phi(exp(-Pi/2)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.

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A003793 Order of (usually) simple Chevalley group A_n (3).

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A132268 Decimal expansion of Product_{k>0} (1-1/12^k).

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A028665 Galois numbers for p=3; order of group AGL(n,3).

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A003790 Order of universal Chevalley group A_n (7).

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A132267 Decimal expansion of Product_{k>0} (1-1/11^k).

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