A132019 -id:A132019 - 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

A025192 a(0)=1; a(n) = 2*3^(n-1) for n >= 1.

Original entry on oeis.org

1, 2, 6, 18, 54, 162, 486, 1458, 4374, 13122, 39366, 118098, 354294, 1062882, 3188646, 9565938, 28697814, 86093442, 258280326, 774840978, 2324522934, 6973568802, 20920706406, 62762119218, 188286357654, 564859072962, 1694577218886, 5083731656658, 15251194969974

Offset: 0

Clark Kimberling

Warning: there is considerable overlap between this entry and the essentially identical A008776.
Shifts one place left when plus-convolved (PLUSCONV) with itself. a(n) = 2*Sum_{i=0..n-1} a(i). - Antti Karttunen, May 15 2001
Let M = { 0, 1, ..., 2^n-1 } be the set of all n-bit numbers. Consider two operations on this set: "sum modulo 2^n" (+) and "bitwise exclusive or" (XOR). The results of these operations are correlated.
To give a numerical measure, consider the equations over M: u = x + y, v = x XOR y and ask for how many pairs (u,v) is there a solution? The answer is exactly a(n) = 2*3^(n-1) for n >= 1. The fraction a(n)/4^n of such pairs vanishes as n goes to infinity. - Max Alekseyev, Feb 26 2003
Number of (s(0), s(1), ..., s(2n+2)) such that 0 < s(i) < 6 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n+2, s(0) = 3, s(2n+2) = 3. - Herbert Kociemba, Jun 10 2004
Number of compositions of n into parts of two kinds. For a string of n objects, before the first, choose first kind or second kind; before each subsequent object, choose continue, first kind, or second kind. For example, compositions of 3 are 3; 2,1; 1,2; and 1,1,1. Using parts of two kinds, these produce respectively 2, 4, 4 and 8 compositions, 2+4+4+8 = 18. - Franklin T. Adams-Watters, Aug 18 2006
In the compositions the kinds of parts are ordered inside a run of identical parts, see example.  Replacing "ordered" by "unordered" gives A052945. - Joerg Arndt, Apr 28 2013
Number of permutations of {1, 2, ..., n+1} such that no term is more than 2 larger than its predecessor. For example, a(3) = 18 because all permutations of {1, 2, 3, 4} are valid except 1423, 1432, 2143, 3142, 2314, 3214, in which 1 is followed by 4. Proof: removing (n + 1) gives a still-valid sequence. For n >= 2, can insert (n + 1) either at the beginning or immediately following n or immediately following (n - 1), but nowhere else. Thus the number of such permutations triples when we increase the sequence length by 1. - Joel B. Lewis, Nov 14 2006
Antidiagonal sums of square array A081277. - Philippe Deléham, Dec 04 2006
Equals row sums of triangle A160760. - Gary W. Adamson, May 25 2009
Let M = a triangle with (1, 2, 4, 8, ...) as the left border and all other columns = (0, 1, 2, 4, 8, ...). A025192 = lim_{n->oo} M^n, the left-shifted vector considered as a sequence. - Gary W. Adamson, Jul 27 2010
Number of nonisomorphic graded posets with 0 and uniform hasse graph of rank n with no 3-element antichain. ("Uniform" used in the sense of Retakh, Serconek and Wilson. By "graded" we mean that all maximal chains have the same length n.) - David Nacin, Feb 13 2012
Equals partial sums of A003946 prefaced with a 1: (1, 1, 4, 12, 36, 108, ...). - Gary W. Adamson, Feb 15 2012
Number of vertices (or sides) of the (n-1)-th iteration of a Gosper island. - Arkadiusz Wesolowski, Feb 07 2013
Row sums of triangle in A035002. - Jon Perry, May 30 2013
a(n) counts walks (closed) on the graph G(1-vertex; 1-loop, 1-loop, 2-loop, 2-loop, 3-loop, 3-loop, ...). - David Neil McGrath, Jan 01 2015
From Tom Copeland, Dec 03 2015: (Start)
For n > 0, a(n) are the traces of the even powers of the adjacency matrix M of the simple Lie algebra B_3, tr(M^(2n)) where M = Matrix(row 1; row 2; row 3) = Matrix[0,1,0; 1,0,2; 0,1,0], same as the traces of Matrix[0,2,0; 1,0,1; 0,1,0] (cf. Damianou). The traces of the odd powers vanish.
The characteristic polynomial of M equals determinant(x*I - M) = x^3 - 3x = A127672(3,x), so 1 - 3*x^2 = det(I - x M) = exp(-Sum_{n>=1} tr(M^n) x^n / n), implying Sum_{n>=1} a(n+1) x^(2n) / (2n) = -log(1 - 3*x^2), giving a logarithmic generating function for the aerated sequence, excluding a(0) and a(1).
a(n+1) = tr(M^(2n)), where tr(M^n) = 3^(n/2) + (-1)^n * 3^(n/2) = 2^n*(cos(Pi/6)^n + cos(5*Pi/6)^n) = n-th power sum of the eigenvalues of M = n-th power sum of the zeros of the characteristic polynomial.
The relation det(I - x M) = exp(-Sum_{n>=1} tr(M^n) x^n / n) = Sum_{n>=0} P_n(-tr(M), -tr(M^2), ..., -tr(M^n)) x^n/n! = exp(P.(-tr(M), -tr(M^2), ...)x), where P_n(x(1), ..., x(n)) are the partition polynomials of A036039 implies that with x(2n) = -tr(M^(2n)) = -a(n+1) for n > 0 and x(n) = 0 otherwise, the partition polynomials evaluate to zero except for P_2(x(1), x(2)) = P_2(0,-6) = -6.
Because of the inverse relation between the partition polynomials of A036039 and the Faber polynomials F_k(b1,b2,...,bk) of A263916, F_k(0,-3,0,0,...) = tr(M^k) gives aerated a(n), excluding n=0,1. E.g., F_2(0,-3) = -2(-3) = 6, F_4(0,-3,0,0) = 2 (-3)^2 = 18, and F_6(0,-3,0,0,0,0) = -2(-3)^3 = 54. (Cf. A265185.)
(End)
Number of permutations of length n > 0 avoiding the partially ordered pattern (POP) {1>2, 1>3, 1>4} of length 4. That is, number of length n permutations having no subsequences of length 4 in which the first element is the largest. - Sergey Kitaev, Dec 08 2020
For n > 0, a(n) is the number of 3-colorings of the grid graph P_2 X P_(n-1). More generally, for q > 1, the number of q-colorings of the grid graph P_2 X P_n is given by q*(q - 1)*((q - 1)*(q - 2) + 1)^(n - 1). - Sela Fried, Sep 25 2023
For n > 1, a(n) is the largest solution to the equation phi(x) = a(n-1). - M. Farrokhi D. G., Oct 25 2023
Number of dotted compositions of degree n. - Diego Arcis, Feb 01 2024

			There are a(3)=18 compositions of 3 into 2 kinds of parts. Here p:s stands for "part p of sort s":
01:  [ 1:0  1:0  1:0  ]
02:  [ 1:0  1:0  1:1  ]
03:  [ 1:0  1:1  1:0  ]
04:  [ 1:0  1:1  1:1  ]
05:  [ 1:0  2:0  ]
06:  [ 1:0  2:1  ]
07:  [ 1:1  1:0  1:0  ]
08:  [ 1:1  1:0  1:1  ]
09:  [ 1:1  1:1  1:0  ]
10:  [ 1:1  1:1  1:1  ]
11:  [ 1:1  2:0  ]
12:  [ 1:1  2:1  ]
13:  [ 2:0  1:0  ]
14:  [ 2:0  1:1  ]
15:  [ 2:1  1:0  ]
16:  [ 2:1  1:1  ]
17:  [ 3:0  ]
18:  [ 3:1  ]
- _Joerg Arndt_, Apr 28 2013
G.f. = 1 + 2*x + 6*x^2 + 18*x^3 + 54*x^4 + 162*x^5 + 486*x^6 + 1458*x^7 + ...

Richard P. Stanley, Enumerative combinatorics, Vol. 1, Cambridge University Press, Cambridge, 1997, pp. 96-100.

T. D. Noe, Table of n, a(n) for n = 0..200
George E. Andrews and Greg Simay, Parity palindrome compositions, Integers 21, Paper No A85, (2021), 13 pp.
Diego Arcis, Camilo González, and Sebastián Márquez On the Hopf algebra of noncommutative symmetric functions in superspace, arXiv:2205.11813 [math.CO], 2022.
D. Bevan, D. Levin, P. Nugent, J. Pantone and L. Pudwell, Pattern avoidance in forests of binary shrubs, arXiv preprint arXiv:1510.08036 [math.CO], 2015.
Fan Chung and R. L. Graham, Primitive juggling sequences, Am. Math. Monthly 115 (3) (2008) 185-194.
Pantelis A. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2014.
Nachum Dershowitz, Between Broadway and the Hudson: A Bijection of Corridor Paths, arXiv:2006.06516 [math.CO], 2020.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 13.
Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
V. Retakh, S. Serconek, and R. Wilson, Hilbert Series of Algebras Associated to Directed Graphs and Order Homology, arXiv:1010.6295 [math.RA], 2010-2011.
Jacob Sprittulla, On Colored Factorizations, arXiv:2008.09984 [math.CO], 2020.
Richard P. Stanley, An Equivalence Relation on the Symmetric Group and Multiplicity-free Flag h-Vectors.
Kai Ting Keshia Yap, David Wehlau, and Imed Zaguia, Permutations Avoiding Certain Partially-ordered Patterns, arXiv:2101.12061 [math.CO], 2021.
Vincent Vatter, Counting parity palindrome compositions, arXiv:2109.13155 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (3).

First differences of 3^n (A000244). Other self-convolved sequences: A000108, A007460, A007461, A007462, A007463, A007464, A061922.
Apart from initial term, same as A008776.
Cf. A132019, A160760, A003946, A035002.
Cf. A036039, A263916, A265185, A127672, A328778.

a025192 0 = 1
a025192 n = 2 * 3 ^ (n -1)
a025192_list = 1 : iterate (* 3) 2  -- Reinhard Zumkeller, Nov 27 2012

A025192 := proc(n): if n=0 then 1 else 2*3^(n-1) fi: end: seq(A025192(n),n=0..26);

Join[{1},2*3^(Range[30]-1)]  (* Harvey P. Dale, Mar 22 2011 *)

a(n)=max(1,2*3^(n-1)) \\ Charles R Greathouse IV, Jul 25 2011

Vec((1-x)/(1-3*x) + O(x^100)) \\ Altug Alkan, Dec 05 2015

[1]+[2*3**(n-1) for n in range(1,30)] # David Nacin, Mar 04 2012

G.f.: (1-x)/(1-3*x).
E.g.f.: (2*exp(3*x) + exp(0))/3. - Paul Barry, Apr 20 2003
a(n) = phi(3^n) = A000010(A000244(n)). - Labos Elemer, Apr 14 2003
a(0) = 1, a(n) = Sum_{k=0..n-1} (a(k) + a(n-k-1)). - Benoit Cloitre, Jun 24 2003
a(n) = A002326((3^n-1)/2). - Vladimir Shevelev, May 26 2008
a(1) = 2, a(n) = 3*a(n-1). - Vincenzo Librandi, Jan 01 2011
a(n) = lcm(a(n-1), Sum_{k=1..n-1} a(k)) for n >= 3. - David W. Wilson, Sep 27 2011
a(n) = ((2*n-1)*a(n-1) + (3*n-6)*a(n-2))/(n-1); a(0)=1, a(1)=2. - Sergei N. Gladkovskii, Jul 16 2012
From Sergei N. Gladkovskii, Jul 17 2012: (Start)
For the e.g.f. E(x) = (2/3)*exp(3*x) + exp(0)/3 we have
E(x) = 2*G(0)/3 where G(k) = 1 + k!/(3*(9*x)^k - 3*(9*x)^(2*k+1)/((9*x)^(k+1) + (k+1)!/G(k+1))); (continued fraction, 3rd kind, 3-step).
E(x) = 1+2*x/(G(0)-3*x) where G(k) = 3*x + 1 + k - 3*x*(k+1)/G(k+1); (continued fraction, Euler's 1st kind, 1-step). (End)
a(n) = A114283(0,0). - Reinhard Zumkeller, Nov 27 2012
G.f.: 1 + ((1/2)/G(0) - 1)/x where G(k) = 1 - 2^k/(2 - 4*x/(2*x - 2^k/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 22 2012
G.f.: 1 + x*W(0), where W(k) = 1 + 1/(1 - x*(2*k+3)/(x*(2*k+4) + 1/W(k+1))); (continued fraction). - Sergei N. Gladkovskii, Aug 28 2013
G.f.: 1 / (1 - 2*x / (1 - x)). - Michael Somos, Apr 03 2014
Construct the power matrix T(n,j) = [A(n)^*j]*[S(n)^*(j-1)] where A(n)=(2,2,2,...) and S(n)=(0,1,0,0,...). (* is convolution operation.) Then a(n) = Sum_{j=1..n} T(n,j). - David Neil McGrath, Jan 01 2015
G.f.: 1 + 2*x/(1 + 2*x)*( 1 + 5*x/(1 + 5*x)*( 1 + 8*x/(1 + 8*x)*( 1 + 11*x/(1 + 11*x)*( 1 + .... - Peter Bala, May 27 2017
Sum_{n>=0} 1/a(n) = 7/4. - Bernard Schott, Oct 02 2021
From Amiram Eldar, May 08 2023: (Start)
Sum_{n>=0} (-1)^n/a(n) = 5/8.
Product_{n>=1} (1 - 1/a(n)) = A132019. (End)

Additional comments from Barry E. Williams, May 27 2000
a(22) corrected by T. D. Noe, Feb 08 2008
Maple programs simplified by Johannes W. Meijer, Jun 02 2011

A067080 If n = ab...def in decimal notation then the left digitorial function Ld(n) = ab...defab...deab...d...ab*a.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 160, 164, 168, 172, 176, 180, 184, 188, 192, 196, 250, 255, 260, 265, 270, 275, 280, 285, 290, 295, 360, 366, 372

Offset: 1

Amarnath Murthy, Jan 05 2002

This entry should probably start at n=0, just as A067079 does. But that would require a number of changes, so it can wait until the editors have more free time. - N. J. A. Sloane, Nov 29 2014

			Ld(256) = 256*25*2 =12800.
a(31)=floor(31/10^0)*floor(31/10^1)=31*3=93;
a(42)=168 since 42=42(base-10) and so a(42)=42*4(base-10)=42*4=168.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Cf. A067079, A065039, A048651, A098844, A132019, A132026, A132038, 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), A132027(p=3)-A132033(p=9), A132263(p=11), A132264(p=12).
For the products of terms 1+floor(n/p^k) see A132269-A132272, A132327, A132328.

a067080 n = if n <= 9 then n else n * a067080 (n `div` 10)
-- Reinhard Zumkeller, Nov 29 2012

Table[d = IntegerDigits[n]; rd = 1; While[ Length[d] > 0, rd = rd*FromDigits[d]; d = Drop[d, -1]]; rd, {n, 1, 75} ]
Table[Times@@NestList[Quotient[#,10]&,n,IntegerLength[n]-1],{n,70}] (* Harvey P. Dale, Dec 16 2013 *)

a(n)=my(t=n);while(n\=10,t*=n); t \\ Charles R Greathouse IV, Nov 20 2012

a(n) = Product_{k=1..length(n)} floor(n/10^(k-1)). - Vladeta Jovovic, Jan 08 2002
From Hieronymus Fischer, Aug 13 2007: (Start)
a(n) = product{0<=k<=floor(log_10(n)), floor(n/10^k)}, n>=1.
Recurrence:
a(n) = n*a(floor(n/10));
a(n*10^m) = n^m*10^(m(m+1)/2)*a(n).
a(k*10^m) = k^(m+1)*10^(m(m+1)/2), for 0a(n) <= b(n), where b(n)=n^(1+floor(log_10(n)))/10^(1/2*(1+floor(log_10(n)))*floor(log_10(n))); equality holds for n=k*10^m, m>=0, 1<=k<10. Here b(n) can also be written n^(1+floor(log_10(n)))/10^A000217(floor(log_10(n))).
Also: a(n) <= 3^((1-log_10(3))/2)*n^((1+log_10(n))/2)=1.332718...*10^A000217(log_10(n)), equality for n=3*10^m, m>=0.
a(n) > c*b(n), where c=0.472362443816572... (see constant A132026).
Also: a(n) > c*2^((1-log_10(2))/2)*n^((1+log_10(n))/2) = 0.601839...*10^A000217(log_10(n)).
lim inf a(n)/b(n) = 0.472362443816572..., for n-->oo.
lim sup a(n)/b(n) = 1, for n-->oo.
lim inf a(n)/n^((1+log_10(n))/2) = 0.472362443816572...*sqrt(2)/2^log_10(sqrt(2)), for n-->oo.
lim sup a(n)/n^((1+log_10(n))/2) = sqrt(3)/3^log_10(sqrt(3)), for n-->oo.
lim inf a(n)/a(n+1) = 0.472362443816572... for n-->oo (see constant A132026).
a(n) = O(n^((1+log_10(n))/2)). (End)

More terms from Robert G. Wilson v, Jan 07 2002

A098844 a(1)=1, a(n) = n*a(floor(n/2)).

Original entry on oeis.org

1, 2, 3, 8, 10, 18, 21, 64, 72, 100, 110, 216, 234, 294, 315, 1024, 1088, 1296, 1368, 2000, 2100, 2420, 2530, 5184, 5400, 6084, 6318, 8232, 8526, 9450, 9765, 32768, 33792, 36992, 38080, 46656, 47952, 51984, 53352, 80000, 82000, 88200, 90300

Offset: 1

Benoit Cloitre, Nov 03 2004

nonn

			a(10) = floor(10/2^0)*floor(10/2^1)*floor(10/2^2)*floor(10/2^3) = 10*5*2*1 = 100;
a(17) = 1088 since 17 = 10001(base 2) and so a(17) = 10001*1000*100*10*1(base 2) = 17*8*4*2*1 = 1088.

Hieronymus Fischer, Table of n, a(n) for n = 1..1000

Cf. A048651, A067080, A132027, A132028, A132029, A132030, A132019, A132026, A132038.
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=3 to p=12 see A132027(p=3)-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.

lst={};Do[p=n;s=1;While[p>1,p=IntegerPart[p/2];s*=p;];AppendTo[lst,s],{n,1,6!,2}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 28 2009 *)

PARI
```
a(n)=if(n<2,1,n*a(floor(n/2)))
```

from math import prod
def A098844(n): return n*prod(n//2**k for k in range(1,n.bit_length()-1)) # Chai Wah Wu, Jun 07 2022

a(2^n) = 2^(n*(n+1)/2) = A006125(n+1).
From Hieronymus Fischer, Aug 13 2007: (Start)
a(n) = Product_{k=0..floor(log_2(n))} floor(n/2^k), n>=1.
Recurrence:
a(n*2^m) = n^m*2^(m(m+1)/2)*a(n).
a(n) <= n^((1+log_2(n))/2) = 2^A000217(log_2(n)); equality iff n is a power of 2.
a(n) >= c(n)*(n+1)^((1 + log_2(n+1))/2) for n != 2,
where c(n) = Product_{k=1..floor(log_2(n))} (1 - 1/2^k); equality holds iff n+1 is a power of 2.
a(n) > c*(n+1)^((1 + log_2(n+1))/2)
where c = 0.288788095086602421... (see constant A048651).
lim inf a(n)/n^((1+log_2(n))/2)=0.288788095086602421... for n-->oo.
lim sup a(n)/n^((1+log_2(n))/2) = 1 for n-->oo.
lim inf a(n)/a(n+1) = 0.288788095086602421... for n-->oo (see constant A048651).
a(n) = O(n^((1+log_2(n))/2)). (End)

Formula section edited by Hieronymus Fischer, Jun 13 2012

A100220 Decimal expansion of Product_{k>=1} (1 - 1/3^k).

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Nov 09 2004

Limit of the probability that a random N X N matrix, with entries chosen independently and uniformly from the field F_3, is nonsingular [Morrison (2006)]. - L. Edson Jeffery, Jan 22 2012

			0.56012607792794894496979224331414001437973633379836...

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.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
Eric Weisstein's World of Mathematics, Infinite Product.

Cf. A048651, A027871.

Cf. A000203, A100221, A100222, A132019, A132034, A132035, A132036, A132037, A132038, A258458.

N[(3^(1/24)*EllipticThetaPrime[1, 0, 1/Sqrt[3]]^(1/3))/2^(1/3)]
N[QPochhammer[1/3,1/3]] (* G. C. Greubel, Nov 27 2015 *)

exp(-Sum_{k > 0} sigma_1(k)/k/3^k) = exp(-Sum_{k > 0} A000203(k)/k/3^k). - Hieronymus Fischer, Aug 07 2007
Product_{k >= 1} (1 - 1/3^k) = (1/3; 1/3){infinity}, where (a;q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Nov 27 2015
From Peter Bala, Jan 18 2021: (Start)
Constant C = (1 - 1/3)*Sum_{n >= 0} (-1/3)^n/Product_{k = 1..n} (3^k - 1);
C = (1 - 1/3)*(1 - 1/9)*Sum_{n >= 0} (-1/9)^n/Product_{k = 1..n} (3^k - 1);
C = (1 - 1/3)*(1 - 1/9)*(1 - 1/27)*Sum_{n >= 0} (-1/27)^n/Product_{k = 1..n} (3^k - 1), and so on. (End)
From Amiram Eldar, Feb 19 2022: (Start)
Equals sqrt(2*Pi/log(3)) * exp(log(3)/24 - Pi^2/(6*log(3))) * Product_{k>=1} (1 - exp(-4*k*Pi^2/log(3))) (McIntosh, 1995).
Equals Sum_{n>=0} (-1)^n/A027871(n). (End)

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)

A132026 Decimal expansion of Product_{k>=0} (1 - 1/(2*10^k)).

Original entry on oeis.org

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

Offset: 0

Hieronymus Fischer, Jul 28 2007

			0.472362443816572236551413383332...

Cf. A000217, A067080, A093136, A098844, A132019.

digits = 103; Product[1-1/(2*10^k), {k, 0, Infinity}] // N[#, digits+1]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
RealDigits[QPochhammer[1/2, 1/10], 10, 100][[1]] (* Jan Mangaldan, Jan 04 2017 *)

prodinf(k=0, 1 - 1/(2*10^k)) \\ Amiram Eldar, May 09 2023

Equals lim inf_{n->oo} Product_{k=0..floor(log_10(n))} floor(n/10^k)*10^k/n.
Equals lim inf_{n->oo} A067080(n)/n^(1+floor(log_10(n)))*10^(1/2*(1+floor(log_10(n)))*floor(log_10(n))).
Equals lim inf_{n->oo} A067080(n)/n^(1+floor(log_10(n)))*10^A000217(floor(log_10(n))).
Equals lim inf_{n->oo} A067080(n)/A067080(n+1).
Equals 1/2*exp(-Sum_{n>0} 10^(-n)*Sum_{k|n} 1/(k*2^k)).
Equals Product_{n>=1} (1 - 1/A093136(n)). - Amiram Eldar, May 09 2023

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)).

A079555 Decimal expansion of Product_{k>=1} (1 + 1/2^k) = 2.384231029031371...

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Jan 25 2003

			2.38423102903137172414989928867839723877161951650843345769...

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. A028362, A048651, A081845, A098844, A100220, A132019-A132026, A132034-A132038, A132265-A132268, A132323-A132326, A132269, A132270, A000593.

Cf. A141848. - Gleb Koloskov, Apr 04 2021

digits = 105; NProduct[(1 + 1/2^k), {k, 1, Infinity}, WorkingPrecision -> digits+10, NProductFactors -> 200] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 14 2013 *)
N[QPochhammer[-1/2,1/2]] (* G. C. Greubel, Dec 05 2015 *)
1/N[QPochhammer[1/2, 1/4]] (* Gleb Koloskov, Apr 04 2021 *)

prodinf(n=1,1+2.^-n) \\ Charles R Greathouse IV, May 27 2015

1/prodinf(n=0, 1-2^(-2*n-1)) \\ Gleb Koloskov, Apr 04 2021

(1/2)*lim sup Product_{k=0..floor(log_2(n)), (1 + 1/floor(n/2^k))} for n-->oo. - Hieronymus Fischer, Aug 20 2007
(1/2)*lim sup A132369(n)/A098844(n) for n-->oo. - Hieronymus Fischer, Aug 20 2007
(1/2)*lim sup A132269(n)/n^((1+log_2(n))/2) for n-->oo. - Hieronymus Fischer, Aug 20 2007
(1/2)*lim sup A132270(n)/n^((log_2(n)-1)/2) for n-->oo. - Hieronymus Fischer, Aug 20 2007
exp(sum{n>0, 2^(-n)*sum{k|n, -(-1)^k/k}})=exp(sum{n>0, A000593(n)/(n*2^n)}). - Hieronymus Fischer, Aug 20 2007
(1/2)*lim sup A132269(n+1)/A132269(n)=2.3842310290313717241498992886... for n-->oo. - Hieronymus Fischer, Aug 20 2007
Equals (-1/2; 1/2){infinity}, where (a;q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Dec 05 2015
2 + Sum_{k>1} 1/(Product_{i=2..k} (2^i-1)) = 2 + 1/3 + 1/(3*7) + 1/(3*7*15) + 1/(3*7*15*31) + 1/(3*7*15*31*63) + ... (conjecture). - Werner Schulte, Dec 22 2016
From Peter Bala, Dec 15 2020: (Start)
The above conjecture of Schulte follows by setting x = 1/2 and t = -1 in the identity Product_{k >= 1} (1 - t*x^k) = Sum_{n >= 0} (-1)^n*x^(n*(n+1)/2)*t^n/( Product_{k = 1..n} 1 - x^k ), due to Euler.
Constant C = 1 + Sum_{n >= 0} (1/2)^(n+1)*Product_{k = 1..n} (1 + 1/2^k).
C = 2 + Sum_{n >= 0} (1/4)^(n+1)*Product_{k = 1..n} (1 + 1/2^k).
3*C = 7 + Sum_{n >= 0} (1/8)^(n+1)*Product_{k = 1..n} (1 + 1/2^k).
3*7*C = 50 + Sum_{n >= 0} (1/16)^(n+1)*Product_{k = 1..n} (1 + 1/2^k).
3*7*15*C = 751 + Sum_{n >= 0} (1/32)^(n+1)*Product_{k = 1..n} (1 + 1/2^k).
(End)
Equals 1/(1-P), where P is the Pell constant from A141848. - Gleb Koloskov, Apr 04 2021
Equals Sum_{k>=0} A000009(k)/2^k. - Vaclav Kotesovec, Sep 15 2021
From Amiram Eldar, Feb 19 2022: (Start)
Equals (sqrt(2)/2) * exp(log(2)/24 + Pi^2/(12*log(2))) * Product_{k>=1} (1 - exp(-2*(2*k-1)*Pi^2/log(2))) (McIntosh, 1995).
Equals (1/2) * A081845.
Equals Sum_{n>=0} 1/A005329(n). (End)

A081845 Decimal expansion of Product_{k>=0} (1 + 1/2^k).

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Apr 09 2003

Twice the product in A079555.

			4.76846205806274344829979857....

G. C. Greubel, Table of n, a(n) for n = 1..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; citeseerx.
József Sándor, On Vandiver's arithmetical function - I, Notes on Number Theory and Discrete Mathematics, Vol. 27, No. 3 (2021), pp. 29-38. See p. 36, eq. 40.

Cf. A006125, A020696, A028361, A079555, A083864.

Cf. A000593, A048651, A100220, A098844, A132019-A132026, A132034-A132038, A132265-A132268, A132323-A132326, A132269, A132270.

digits = 105; NProduct[1 + 1/2^k, {k, 0, Infinity}, WorkingPrecision -> digits+5, NProductFactors -> digits] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Mar 04 2013 *)
N[QPochhammer[-1, 1/2], 100] (* Vaclav Kotesovec, Dec 13 2015 *)
2*N[QPochhammer[-1/2, 1/2], 200] (* G. C. Greubel, Dec 20 2015 *)

prodinf(k=0,1/2^k,1) \\ Hugo Pfoertner, Feb 21 2020

lim sup Product_{k=0..floor(log_2(n))} (1 + 1/floor(n/2^k)) for n-->oo. - Hieronymus Fischer, Aug 20 2007
lim sup A132369(n)/A098844(n) for n-->oo. - Hieronymus Fischer, Aug 20 2007
lim sup A132269(n)/n^((1+log_2(n))/2) for n-->oo. - Hieronymus Fischer, Aug 20 2007
lim sup A132270(n)/n^((log_2(n)-1)/2) for n-->oo. - Hieronymus Fischer, Aug 20 2007
2*exp(Sum_{n>0} 2^(-n)*Sum_{k|n} -(-1)^k/k) = 2*exp(Sum_{n>0} A000593(n)/(n*2^n)). - Hieronymus Fischer, Aug 20 2007
lim sup A132269(n+1)/A132269(n) = 4.76846205806274344... for n-->oo. - Hieronymus Fischer, Aug 20 2007
Sum_{k>=1} (-1)^(k+1) * 2^k / (k*(2^k-1)) = log(A081845) = 1.562023833218500307570359922772014353168080202860122... . - Vaclav Kotesovec, Dec 13 2015
Equals 2*(-1/2; 1/2){infinity}, where (a;q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Dec 20 2015
Equals 1 + Sum_{n>=1} 2^n/((2-1)*(2^2-1)*...*(2^n-1)). - Robert FERREOL, Feb 21 2020
From Peter Bala, Jan 18 2021: (Start)
Constant C = 3*Sum_{n >= 0} (1/2)^n/Product_{k = 1..n} (2^k - 1).
Faster converging series:
C = (2*3*5)/(2^3)*Sum_{n >= 0} (1/4)^n/Product_{k = 1..n} (2^k - 1),
C = (2*3*5*9)/(2^6)*Sum_{n >= 0} (1/8)^n/Product_{k = 1..n} (2^k - 1),
C = (2*3*5*9*17)/(2^10)*Sum_{n >= 0} (1/16)^n/Product_{k = 1..n} (2^k - 1), and so on. The sequence [2,3,5,9,17,...] is A000051. (End)
From Amiram Eldar, Mar 20 2022: (Start)
Equals sqrt(2) * exp(log(2)/24 + Pi^2/(12*log(2))) * Product_{k>=1} (1 - exp(-2*(2*k-1)*Pi^2/log(2))) (McIntosh, 1995).
Equals 1/A083864. (End)
Equals lim_{n->oo} A020696(2^n)/A006125(n+1) (Sándor, 2021). - Amiram Eldar, Jun 29 2022

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

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A025192 a(0)=1; a(n) = 2*3^(n-1) for n >= 1.

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A067080 If n = ab...def in decimal notation then the left digitorial function Ld(n) = ab...def*ab...de*ab...d*...*ab*a.

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A098844 a(1)=1, a(n) = n*a(floor(n/2)).

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A100220 Decimal expansion of Product_{k>=1} (1 - 1/3^k).

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

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A067080 If n = ab...def in decimal notation then the left digitorial function Ld(n) = ab...defab...deab...d...ab*a.