A335764 -id:A335764 - 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

A066524 a(n) = n*(2^n - 1).

Original entry on oeis.org

0, 1, 6, 21, 60, 155, 378, 889, 2040, 4599, 10230, 22517, 49140, 106483, 229362, 491505, 1048560, 2228207, 4718574, 9961453, 20971500, 44040171, 92274666, 192937961, 402653160, 838860775, 1744830438, 3623878629, 7516192740, 15569256419, 32212254690

Offset: 0

Henry Bottomley, Jan 08 2002

a(n)/2^n is the expected value of the cardinality of the generalized union of n randomly selected (with replacement) subsets of [n] where the probability of selection is equal for all subsets. - Geoffrey Critzer, May 18 2009

Form a triangle in which interior members T(i,j) = T(i-1,j-1) + T(i-1,j). The exterior members are given by 1,2,3,...,2*n-1: T(1,1) = n, T(2,1) = n-1, T(3,1) = n-2, ..., T(n,1) = 1 and T(2,2) = n + 1, T(3,3) = n + 2, ..., T(n,n) = 2*n - 1. The sum of all members will reproduce this sequence. For example, with n = 4 the exterior members are 1 to 7: row(1) = 4; row(2) = 3,5; row(3) = 2,8,6; row(4) = 1,10,14,7. The sum of all these members is 60, the fourth term in the sequence. - J. M. Bergot, Oct 16 2012

			a(4) = 4*(2^4 - 1) = 4*15 = 60.

Harry J. Smith, Table of n, a(n) for n = 0..250
A. Umar, Combinatorial Results for Semigroups of Orientation-Preserving Partial Transformations, Journal of Integer Sequences, 14 (2011), #11.7.5.
Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

Cf. A000225, A132751, A335764.

[n*(2^n-1): n in [0..30]]; // Vincenzo Librandi, Jan 24 2016

Table[n*2^n-n,{n,0,3*4!}] (* Vladimir Joseph Stephan Orlovsky, Apr 25 2010 *)
CoefficientList[Series[x (1 - 2 x^2)/((1 - x) (1 - 2 x))^2, {x, 0, 30}], x] (* Michael De Vlieger, Jan 24 2016 *)

[gaussian_binomial(n,1,2)*n for n in range(0,29)] # Zerinvary Lajos, May 29 2009

a(n) = 2*a(n-1) + 2^n = A000225(n) * A001477(n) = A036289(n) - A001477(n).
G.f.: x*(1 - 2*x^2)/((1 - x)*(1 - 2*x))^2.
a(n) = n * Sum_{j = 1..n} binomial(n,j), n >= 0. - Zerinvary Lajos, May 10 2007
Row sums of triangles A132751. - Gary W. Adamson, Aug 28 2007
E.g.f.: x*(2*exp(2*x) - exp(x)). From an earlier rewritten comment. - Wolfdieter Lang, Feb 16 2016
Sum_{n>=1} 1/a(n) = A335764. - Amiram Eldar, Jun 23 2020

A335763 Decimal expansion of Sum_{k>=1} sigma_2(k)/2^k where sigma_2(k) is the sum of squares of divisors of k (A001157).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 21 2020

			7.099285178890907114033125022164753663157608833211895...

Maxie Dion Schmidt, A catalog of interesting and useful Lambert series identities, arXiv:2004.02976 [math.NT], 2020.
Eric Weisstein's World of Mathematics, Lambert Series.
Wikipedia, Lambert series.

Cf. A001157, A065442, A066766, A227989, A256936, A335764.

evalf(add( (1/2)^(n^2)*( n^2*(8^n) - ((n-1)^2 - 2)*4^n - ((n+1)^2 - 2)*(2^n) + n^2 )/(2^n - 1)^3, n = 1..20 ), 100); # Peter Bala, Jan 22 2021

RealDigits[Sum[n^2/(2^n - 1), {n, 1, 500}], 10, 100][[1]]

Equals Sum_{k>=1} k^2/(2^k - 1).

Faster converging series: Sum_{n >= 1} (1/2)^(n^2)*( n^2*(8^n) - ((n-1)^2 - 2)*4^n - ((n+1)^2 - 2)*(2^n) + n^2 )/(2^n - 1)^3. - Peter Bala, Jan 19 2021

A335765 Decimal expansion of Sum_{k>=1} 1/2^(k-omega(k)) where omega(k) is the number of distinct primes dividing k (A001221).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 21 2020

			1.533885641474380666826406030970632881500707940362154...

Maxie Dion Schmidt, A catalog of interesting and useful Lambert series identities, arXiv:2004.02976 [math.NT], 2020.
Eric Weisstein's World of Mathematics, Lambert Series.
Wikipedia, Lambert series.

Cf. A001221, A000225, A005117, A008683, A034444, A065442, A066766, A256936, A335763, A335764.

RealDigits[Sum[1/2^(n - PrimeNu[n]), {n, 1, 500}], 10, 100][[1]]

Equals Sum_{k>=1} A034444(k)/2^k.

Equals Sum_{k>=1} mu(k)^2/(2^k - 1), where mu(k) is the Möbius function (A008683), or, equivalently, Sum_{k>=1} 1/A000225(A005117(k)).

A367325 Decimal expansion of Sum_{k>=1} sigma(k)/(2^k-1), where sigma(k) is the sum of divisors of k (A000203).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Nov 14 2023

			3.60444734197194674489364847362358835600495487064998...

Maxie Dion Schmidt, A catalog of interesting and useful Lambert series identities, arXiv:2004.02976 [math.NT], 2020. See eq. (6.1c), p. 13.
Michael I. Shamos, Shamos's catalog of the real numbers, 2011. See p. 526.

Cf. A000005, A000203.

Similar constants: A065442, A066766, A116217, A335763, A335764.

with(numtheory): evalf(sum(sigma(k)/(2^k-1), k = 1..infinity), 120)

RealDigits[Sum[DivisorSigma[1,n]/(2^n-1), {n, 1, 500}], 10, 120][[1]]

PARI
```
suminf(k = 1, sigma(k)/(2^k-1))
```

suminf(k = 1, numdiv(k)/((2^k-1)*(1-1/2^k)))

Equals Sum_{k>=1} d(k)/((2^k-1)*(1-1/2^k)), where d(k) is the number of divisors of k (A000005).

cp's OEIS Frontend

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

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A066524 a(n) = n*(2^n - 1).

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A335763 Decimal expansion of Sum_{k>=1} sigma_2(k)/2^k where sigma_2(k) is the sum of squares of divisors of k (A001157).

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A335765 Decimal expansion of Sum_{k>=1} 1/2^(k-omega(k)) where omega(k) is the number of distinct primes dividing k (A001221).

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A367325 Decimal expansion of Sum_{k>=1} sigma(k)/(2^k-1), where sigma(k) is the sum of divisors of k (A000203).

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