A065442 -id:A065442 - cp's OEIS frontend

A248725 Decimal expansion of Sum_{k>=1} 1/(8^k - 1).

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

Offset: 0

Robert G. Wilson v, Oct 12 2014

			0.16096618431506239680530256414364288555074385602532834636083591864782394085800...

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A000005, A065442, A073668, A214369, A248721, A248722, A248723, A248724, A248726.

evalf(sum(1/(8^k-1), k=1..infinity),120) # Vaclav Kotesovec, Oct 18 2014
# second program with faster converging series
evalf( add( (1/8)^(n^2)*(1 + 2/(8^n - 1)), n = 1..10), 105); # Peter Bala, Jan 30 2022

x = 1/8; RealDigits[ Sum[ DivisorSigma[0, k] x^k, {k, 1000}], 10, 105][[1]] (* after an observation and the formula of Amarnath Murthy, see A073668 *)

suminf(k=1, 1/(8^k-1)) \\ Michel Marcus, Oct 18 2014

Equals Sum_{k>=1} d(k)/8^k, where d(k) is the number of divisors of k (A000005). - Amiram Eldar, Jun 22 2020

A248726 Decimal expansion of Sum_{k>=1} 1/(9^k - 1).

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v, Oct 12 2014

			0.13904511766218812935872847436908905213936264706781960955103549347967020145366...

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A000005, A065442, A073668, A214369, A248721, A248722, A248723, A248724, A248725.

evalf(sum(1/(9^k-1), k=1..infinity),120) # Vaclav Kotesovec, Oct 18 2014
# second program with faster converging series
evalf( add( (1/9)^(n^2)*(1 + 2/(9^n - 1)), n = 1..10), 105); # Peter Bala, Jan 30 2022

x = 1/9; RealDigits[ Sum[ DivisorSigma[0, k] x^k, {k, 1000}], 10, 105][[1]] (* after an observation and the formula of Amarnath Murthy, see A073668 *)

suminf(k=1, 1/(9^k-1)) \\ Michel Marcus, Oct 18 2014

Equals Sum_{k>=1} d(k)/9^k, where d(k) is the number of divisors of k (A000005). - Amiram Eldar, Jun 22 2020

A066766 Decimal expansion of Sum_{k>=1} sigma(k)/2^k where sigma(k) is the sum of divisors of k, 1 <= d <= k.

Original entry on oeis.org

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

Offset: 1

Randall L Rathbun, Jan 16 2002

			2.74403388875948836048021489149227216431142898131963931784...

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

Thomas Bloom, Is Sum sigma(n)/2^n irrational? (Here sigma(n) is the sum of divisors function), Erdős Problems.
Steven R. Finch, Digital Search Tree Constants [Broken link]
Steven R. Finch, Digital Search Tree Constants [From the Wayback machine]
Terence Tao, Erdős problem database, see no. 250.

Cf. A000203, A065442, A065443, A066772, A335763.

evalf( add( (1/2)^(n^2) * (n*(4^n - 1) + 2^n)/(2^n - 1)^2, n = 1..20), 100); # Peter Bala, Jan 19 2021

RealDigits[Sum[n/(2^n - 1), {n, 1, 500}], 10, 100][[1]] (* Amiram Eldar, Jun 22 2020 *)

smv(v)= s=0; for(i=1,matsize(v)[2],s=s+v[i]); s
A066766(n)= sm=0; for(j=1,n,sm=sm+smv(divisors(j)/2^j)); sm*1.0

suminf(k=1, sigma(k)/2^k) \\ Michel Marcus, Apr 27 2018

Equals Sum_{k>=1} k/(2^k - 1). - Amiram Eldar, Jun 22 2020
Faster converging series: Sum_{n >= 1} (1/2)^(n^2) * (n*(4^n - 1) + 2^n)/(2^n - 1)^2. - Peter Bala, Jan 19 2021
From Amiram Eldar, Oct 16 2022: (Start)
Equals Sum_{k>=1} 2^k/(2^k - 1)^2.
Equals A065442 + A065443. (End)

Name corrected by Paul D. Hanna, Apr 26 2018

A088837 Numerator of sigma(2*n)/sigma(n). Denominator see in A038712.

Original entry on oeis.org

3, 7, 3, 15, 3, 7, 3, 31, 3, 7, 3, 15, 3, 7, 3, 63, 3, 7, 3, 15, 3, 7, 3, 31, 3, 7, 3, 15, 3, 7, 3, 127, 3, 7, 3, 15, 3, 7, 3, 31, 3, 7, 3, 15, 3, 7, 3, 63, 3, 7, 3, 15, 3, 7, 3, 31, 3, 7, 3, 15, 3, 7, 3, 255, 3, 7, 3, 15, 3, 7, 3, 31, 3, 7, 3, 15, 3, 7, 3, 63, 3, 7, 3, 15, 3, 7, 3, 31, 3, 7, 3, 15, 3

Offset: 1

Labos Elemer, Nov 04 2003

In general sigma(2^k*n) / sigma(n) = ((2^k*n) XOR (2^k*n-1)) / (n XOR (n-1)), see link. Jon Maiga, Dec 10 2018

Antti Karttunen, Table of n, a(n) for n = 1..16384
Jon Maiga, Efficient computation of ratios between divisor sums, 2018.
Index entries for sequences related to sigma(n).

Cf. A000203, A038712 (denominators), A062731, A088838, A088839, A088840, A080278, A220466.

Cf. A001620, A065442.

nmax:=93: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 1 to ceil(nmax/(p+2)) do a((2*n-1)*2^p) := 2^(p+2)-1 od: od: seq(a(n), n=1..nmax); # Johannes W. Meijer, Feb 09 2013

k=2; Table[Numerator[DivisorSigma[1, k*n]/DivisorSigma[1, n]], {n, 1, 128}]
Table[BitXor[2*n, 2*n - 1], {n, 128}] (* Jon Maiga, Dec 10 2018 *)

A088837(n) = numerator(sigma(n<<1)/sigma(n)); \\ Antti Karttunen, Nov 01 2018

a(n) = 4*2^A007814(n)-1 = 4*A006519(n)-1 = A059159(n)-1 = 2*A038712(n) + 1.
a((2*n-1)*2^p) = 2^(p+2)-1, p >= 0 and n >= 1. - Johannes W. Meijer, Feb 09 2013
a(n) = (2n) XOR (2n-1). - Jon Maiga, Dec 10 2018
From Amiram Eldar, Jan 06 2023: (Start)
a(n) = numerator(A062731(n)/A000203(n)).
Sum_{k=1..n} a(k) ~ (log_2(n) + (gamma-1)/log(2) + 1)*2*n, where gamma is Euler's constant (A001620).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A038712(k) = A065442 + 1 = 2.606695... . (End).

A211705 Decimal expansion of Sum_{n>=1} A006218(n)*2^(-n).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 19 2012

			Sum_{n>=1} A006218(n)*2^(-n) = 1/2 + 3/4 + 5/8 + 8/16 + 10/32 + 14/64 + ... = 3.213390304830583527566603046381849...

Cf. A006218, A065442, A211701, A211706 (binary).

f[n_, m_] := Sum[Floor[n/k], {k, 1, m}]
t = Table[f[n, 100], {n, 1, 4000}] ;
N[Sum[t[[n]]/2^n, {n, 1, 4000}], 100]
RealDigits[%, 10]  (* A211705 *)
RealDigits[%%, 2]  (* A211706 *)

k=1.; 2*suminf(n=1, k>>=1; k^n*(1+k)/(1-k)) \\ Charles R Greathouse IV, Jul 18 2021

Equals 2 * A065442. - Amiram Eldar, Aug 01 2020

a(31) onward corrected by Sean A. Irvine, Jun 09 2024

A323482 Decimal expansion of 1/2 + 1/3 + 1/5 + ... + 1/(2^n + 1) + ...

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jan 15 2019

			1.2644997803484442091913...

Jinyuan Wang, Table of n, a(n) for n = 1..1000

Cf. A000051, A065442, A008683, A179951.

evalf((1/2) + add( (-1)^(n+1)*((4^n + 1)/(4^n - 1))*(1/2)^(n^2), n = 1..18), 100); # Peter Bala, Jan 28 2022

s = Sum[1/(2^k + 1), {k, 0, Infinity}]
r = N[Re[s], 200]
RealDigits[r][[1]]

suminf(k=0, 1/(2^k+1)) \\ Michel Marcus, Jan 15 2019

From Amiram Eldar, Jun 30 2020: (Start)
Equals 1/2 + Sum_{k>=1} (-1)^(k+1)/(2^k-1)
Equals Sum_{k>=1} (mu(k) - (-1)^k)/(2^k-1), where mu is the Möbius function (A008683).
Equals (1 + A179951)/2. (End)
Equals (1/2) + Sum_{n >= 1} (-1)^(n+1)*((4^n + 1)/(4^n - 1))*(1/2)^(n^2). The first 18 terms of the series gives the constant correct to more than 100 decimal places. - Peter Bala, Jan 28 2022

A088839 Numerator of sigma(4n)/sigma(n).

Original entry on oeis.org

7, 5, 7, 31, 7, 5, 7, 21, 7, 5, 7, 31, 7, 5, 7, 127, 7, 5, 7, 31, 7, 5, 7, 21, 7, 5, 7, 31, 7, 5, 7, 85, 7, 5, 7, 31, 7, 5, 7, 21, 7, 5, 7, 31, 7, 5, 7, 127, 7, 5, 7, 31, 7, 5, 7, 21, 7, 5, 7, 31, 7, 5, 7, 511, 7, 5, 7, 31, 7, 5, 7, 21, 7, 5, 7, 31, 7, 5, 7, 127, 7, 5, 7, 31, 7, 5, 7, 21, 7, 5, 7, 31

Offset: 1

Labos Elemer, Nov 04 2003

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sigma(n).

For denominator see A088840.
Cf. A000203, A038712, A088837, A088838, A088841, A080278, A193553.
Cf. A006519, A065442, A096268.

f:= proc(n) local m;
  m:= padic:-ordp(n,2);
  if m::odd then (2^(m+3)-1)/3 else 2^(m+3)-1 fi
end proc:
map(f, [$1..200]); # Robert Israel, Nov 19 2017

k=4; Table[Numerator[DivisorSigma[1, k*n]/DivisorSigma[1, n]], {n, 1, 128}]

A088839(n) = numerator(sigma(4*n)/sigma(n)); \\ Antti Karttunen, Nov 18 2017

a(n) = (8*A006519(n)-1)/(1+2*A096268(n)). - Robert Israel, Nov 19 2017
From Amiram Eldar, Jan 06 2023: (Start)
a(n) = numerator(A193553(n)/A000203(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A088840(k) = 3*A065442 + 1 = 5.820085... . (End)

Typo in definition corrected by Antti Karttunen, Nov 18 2017

A203011 (n-1)-st elementary symmetric function of {1,3,7,15,31,63,...-1+2^n}.

Original entry on oeis.org

1, 4, 31, 486, 15381, 978768, 124918731, 31932406170, 16337382642945, 16723323142761060, 34243057328337866295, 140246638967945496322350, 1148847521944847479468879725, 18822284044001939139425413111800, 616761496621711735518439444437389475

Offset: 1

Clark Kimberling, Dec 29 2011

nonn

Clark Kimberling, Table of n, a(n) for n = 1..50

Cf. A122743.

SymmPolyn := proc(L::list,n::integer)
    local c,a,sel;
    a :=0 ;
    sel := combinat[choose](nops(L),n) ;
    for c in sel do
        a := a+mul(L[e],e=c) ;
    end do:
    a;
end proc:
A203011 := proc(n)
    local L,k ;
    L := [seq(2^k-1,k=1..n)] ;
    SymmPolyn(L,n-1) ;
end proc: # R. J. Mathar, Sep 23 2016

f[k_] := -1 + 2^k; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 16}] (* A203011 *)
Table[Product[2^k-1,{k,1,n}] * Sum[1/(2^k-1),{k,1,n}],{n,1,16}] (* Vaclav Kotesovec, Sep 06 2014 *)

a(n) = c * 2^(n*(n+1)/2), where c = A048651 * A065442 = 0.4639944324508904477884... . - Vaclav Kotesovec, Oct 10 2016

A086341 a(n) = 2*2^floor(n/2) - (-1)^n.

Original entry on oeis.org

1, 3, 3, 5, 7, 9, 15, 17, 31, 33, 63, 65, 127, 129, 255, 257, 511, 513, 1023, 1025, 2047, 2049, 4095, 4097, 8191, 8193, 16383, 16385, 32767, 32769, 65535, 65537, 131071, 131073, 262143, 262145, 524287, 524289, 1048575, 1048577, 2097151, 2097153

Offset: 0

Paul Barry, Jul 16 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (-1,2,2).

Cf. A016116 (2^floor(n/2)).

Cf. A065442, A248721, A323482.

[2*2^Floor(n/2)-(-1)^n: n in [0..40]]; // Vincenzo Librandi, Aug 16 2011

CoefficientList[Series[(1+2x)^2/((1+x)(1-2x^2)),{x,0,50}],x] (* or *) LinearRecurrence[ {-1,2,2},{1,3,3},50] (* Harvey P. Dale, Mar 10 2013 *)

vector(40, n, n--; 2^(floor(n/2)+1) - (-1)^n) \\ G. C. Greubel, Nov 08 2018

E.g.f.: 2*cosh(sqrt(2)*x) + 2*sinh(sqrt(2)*x)/sqrt(2) - sinh(x) + cosh(x).
a(n) = (1 + 1/sqrt(2))*sqrt(2)^n + (1 - 1/sqrt(2))*(-sqrt(2))^n - (-1)^n.
G.f.: (1+2*x)^2/((1+x)*(1-2*x^2)). - Colin Barker, Aug 17 2012
a(n) = a(n-1) + 2*a(n-2) + 2*a(n-3); a(0)=1, a(1)=3, a(2)=3. - Harvey P. Dale, Mar 10 2013
From Amiram Eldar, Sep 14 2022: (Start)
Sum_{n>=0} 1/a(n) = A065442 + A323482 - 1/2.
Sum_{n>=0} (-1)^n/a(n) = 2 * A248721. (End)

A179951 Decimal expansion of Sum_{k has exactly two bits equal to 1 in base 2} 1/k.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Aug 03 2010

Obviously for k > 0 in base 2 having no bit equal to 1 the sum is 0 and for 1 bit equal to 1 the sum is 2.

			Sum_{k>0} 1/A018900(k) = 1.52899956069688841838263949451...

Robert Baillie, Summing The Curious Series Of Kempner and Irwin, arXiv:0806.4410 [math.CA], 2008-2015.
Wolfram Library Archive, KempnerSums.nb (8.6 KB) - Mathematica Notebook, Summing Kempner's Curious (Slowly-Convergent) Series

Cf. A065442, A082830, A082831, A082832, A082833, A082834, A082835, A082836, A082837, A082838, A082839, A140502, A160502, A018900, A323482.

 evalf( 2*add( (-1)^(n+1)*((4^n + 1)/(4^n - 1))*(1/2)^(n^2), n = 1..18), 100); # Peter Bala, Jan 28 2022

(* first install irwinSums.m, see either reference, then *) First@ RealDigits@ iSum[1, 2, 2^7, 2]

Equals Sum_{j>=1} Sum_{i=0..j-1} 1/(2^i + 2^j).
From Amiram Eldar, Jun 30 2020: (Start)
Equals Sum_{k>=0} 1/(2^k + 1/2).
Equals 2 * A323482 - 1. (End)
Equals 2*Sum_{n >= 1} (-1)^(n+1)*((4^n + 1)/(4^n - 1))*(1/2)^(n^2). The first 18 terms of the series gives the constant correct to more than 100 decimal places. - Peter Bala, Jan 28 2022

cp's OEIS Frontend

A248725 Decimal expansion of Sum_{k>=1} 1/(8^k - 1).

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A248726 Decimal expansion of Sum_{k>=1} 1/(9^k - 1).

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A066766 Decimal expansion of Sum_{k>=1} sigma(k)/2^k where sigma(k) is the sum of divisors of k, 1 <= d <= k.

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A088837 Numerator of sigma(2*n)/sigma(n). Denominator see in A038712.

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A211705 Decimal expansion of Sum_{n>=1} A006218(n)*2^(-n).

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A323482 Decimal expansion of 1/2 + 1/3 + 1/5 + ... + 1/(2^n + 1) + ...

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A088839 Numerator of sigma(4n)/sigma(n).

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