A065442 -id:A065442 - cp's OEIS frontend

A214369 Decimal expansion of Sum_{n>=1} 1/(3^n-1).

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

Offset: 0

R. J. Mathar, Jul 14 2012

			Equals 0.6821535026052380667...

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

Cf. A024023, A065442, A073668, A248721, A248722, A248723, A248724, A248725, A248726.

evalf(sum(1/(3^k-1), k=1..infinity), 120); # Vaclav Kotesovec, Oct 18 2014
# second program with faster converging series
evalf( add( (1/3)^(n^2)*(1 + 2/(3^n - 1)), n = 1..14 ), 105); # Peter Bala, Jan 30 2022

RealDigits[ NSum[1/(3^n - 1), {n, 1, Infinity}, WorkingPrecision -> 110, NSumTerms -> 100], 10, 105] // First (* or *) 1 - (Log[2] + QPolyGamma[0, 1, 1/3])/Log[3] // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Jun 05 2013 *)
x = 1/3; RealDigits[ Sum[ DivisorSigma[0, k] x^k, {k, 1000}], 10, 105][[1]] (* Robert G. Wilson v, Oct 12 2014 after an observation and the formula of Amarnath Murthy, see A073668 *)

suminf(n=1, 1/(3^n-1)) \\ Michel Marcus, Mar 11 2017

Equals Sum_{n>=1} 1/A024023(n).

Equals Sum_{k>=1} d(k)/3^k, where d(k) is the number of divisors of k (A000005). - Amiram Eldar, May 17 2020

More terms from Jean-François Alcover, Feb 12 2013

A248721 Decimal expansion of Sum_{k>=1} 1/(4^k - 1).

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v, Oct 12 2014

			0.4210976860334237772959908879677130489614413363241154046059207967127713704887...

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

Cf. A000005, A065442, A073668, A214369, A248722, A248723, A248724, A248725, A248726.

evalf(sum(1/(4^k-1), k=1..infinity),120) # Vaclav Kotesovec, Oct 18 2014
# second program with faster converging series after Joerg Arndt
evalf( add( (1/4)^(n^2)*(1 + 2/(4^n - 1)), n = 1..13), 105); # Peter Bala, Jan 30 2022

x = 1/4; 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/(4^k-1)) \\ Michel Marcus, Oct 18 2014

Equals Sum_{k>=1} x^(k^2)*(1+x^k)/(1-x^k) where x = 1/4 (the Lambert series evaluated at 1/4). - Joerg Arndt, Jun 03 2020

Equals Sum_{k>=1} d(k)/4^k, where d(k) is the number of divisors of k (A000005). - Amiram Eldar, Jun 22 2020

A065443 Decimal expansion of Sum_{k=1..inf} 1/(2^k-1)^2.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Nov 18 2001

			1.1373387363441965966969133683013475838308493098...

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

Harry J. Smith, Table of n, a(n) for n=1..2000
Steven R. Finch, Digital Search Tree Constants. [Broken link]
Steven R. Finch, Digital Search Tree Constants. [From the Wayback machine]
Eric Weisstein's World of Mathematics, Tree Searching.

Cf. A060867, A065442, A065608, A066766, A067616.

RealDigits[NSum[1/(2^k - 1)^2, {k, 1, Infinity}, PrecisionGoal -> 40, AccuracyGoal -> 40, WorkingPrecision -> 500, NSumTerms -> 50, NSumExtraTerms -> 50]][[1]] (* Peter Bertok (peter(AT)bertok.com), Dec 04 2001 *)
RealDigits[(Log[2] QPolyGamma[0, 1, 1/2] + QPolyGamma[1, 1, 1/2])/Log[2]^2 - 1, 10, 20][[1]] (* Eric W. Weisstein, Jun 02 2025 *)

{ default(realprecision, 2080); x=suminf(k=1, 1/(2^k - 1)^2); for (n=1, 2000, d=floor(x); x=(x-d)*10; write("b065443.txt", n, " ", d)) } \\ Harry J. Smith, Oct 19 2009

Equals Sum_{n>=1} 1/A060867(n).
From Amiram Eldar, Oct 16 2022: (Start)
Equals Sum_{k>=1} k/(2^(k+1)-1).
Equals A066766 - A065442. (End)
Equals Sum_{n >= 1} q^(n^2)*( (n - 1) + q^n - (n - 1)*q^(2*n) )/(1 - q^n)^2 evaluated at q = 1/2 (see A065608). - Peter Bala, Oct 16 2022

More terms from Peter Bertok (peter(AT)bertok.com), Dec 04 2001

A173898 Decimal expansion of sum of the reciprocals of the Mersenne primes.

Original entry on oeis.org

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

Offset: 0

Jonathan Vos Post, Mar 01 2010

We know this a priori to be strictly less than the Erdős-Borwein constant (A065442), which Erdős (1948) showed to be irrational. This new constant would also seem to be irrational.

			Decimal expansion of (1/3) + (1/7) + (1/31) + (1/127) + (1/8191) + (1/131071) + (1/524287) + ... = .5164541789407885653304873429715228588159685534154197.
This has continued fraction expansion 0 + 1/(1 + 1/(1 + 1/(14 + 1/(1 + ...)))) (see A209601).

Peter B. Borwein, On the Irrationality of Certain Series, Math. Proc. Cambridge Philos. Soc. 112, 141-146, 1992.
Paul Erdős, On Arithmetical Properties of Lambert Series, J. Indian Math. Soc. 12, 63-66, 1948.
Yoshihiro Tanaka, On the Sum of Reciprocals of Mersenne Primes, American Journal of Computational Mathematics, Vol. 7, No. 2 (2017), pp. 145-148.
Eric Weisstein's World of Mathematics, Erdos-Borwein Constant.
Marek Wolf, Computer experiments with Mersenne primes, arXiv preprint arXiv:1112.2412 [math.NT], 2011.

Cf. A209601, A000668, A065442 (decimal expansion of Erdos-Borwein constant), A000043, A001348, A046051, A057951-A057958, A034876, A124477, A135659, A019279, A061652, A000225.

Digits := 120 ; L := [ 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917 ] ;
x := 0 ; for i from 1 to 30 do x := x+1.0/(2^op(i,L)-1 ); end do ;

RealDigits[Sum[1/(2^p - 1), {p, MersennePrimeExponent[Range[14]]}], 10, 100][[1]] (* Amiram Eldar, May 24 2020 *)

isM(p)=my(m=Mod(4,2^p-1));for(i=1,p-2,m=m^2-2);!m
s=1/3;forprime(p=3,default(realprecision)*log(10)\log(2), if(isM(p), s+=1./(2^p-1)));s \\ Charles R Greathouse IV, Mar 22 2012

Sum_{i>=1} 1/A000668(i).

Entry revised by N. J. A. Sloane, Mar 10 2012

A240736 Number of compositions of n having exactly one fixed point.

Original entry on oeis.org

1, 1, 1, 4, 7, 16, 29, 60, 120, 238, 479, 956, 1910, 3817, 7633, 15252, 30491, 60955, 121865, 243650, 487165, 974112, 1947851, 3895086, 7789153, 15576624, 31150481, 62296424, 124585395, 249158607, 498297297, 996562085, 1993071152, 3986055928, 7971971230

Offset: 1

Joerg Arndt and Alois P. Heinz, Apr 11 2014

nonn

			a(4) = 4: 13, 22, 112, 1111.
a(5) = 7: 14, 32, 131, 221, 1112, 1121, 11111.

M. Archibald, A. Blecher and A. Knopfmacher, Fixed points in compositions and words, accepted by the Journal of Integer Sequences.

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 1..1000
M. Archibald, A. Blecher, and A. Knopfmacher, Fixed Points in Compositions and Words, J. Int. Seq., Vol. 23 (2020), Article 20.11.1.

Column k=1 of A238349 and of A238350.

b:= proc(n, i) option remember; `if`(n=0, 1, series(
      add(b(n-j, i+1)*`if`(i=j, x, 1), j=1..n), x, 2))
    end:
a:= n-> coeff(b(n, 1), x, 1):
seq(a(n), n=1..40);

b[n_, i_] := b[n, i] = If[n == 0, 1, Series[Sum[b[n - j, i + 1]*If[i == j, x, 1], {j, 1, n}], {x, 0, 2}]]; a[n_] := SeriesCoefficient[b[n, 1], {x, 0, 1}]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Nov 06 2014, after Maple *)

a(n) ~ c * 2^n, where c = A065442 * A048651 / 2 = 0.2319972162254452238942023675457837005318389885... - Vaclav Kotesovec, Sep 06 2014

A016152 a(n) = 4^(n-1)*(2^n-1).

Original entry on oeis.org

0, 1, 12, 112, 960, 7936, 64512, 520192, 4177920, 33488896, 268173312, 2146435072, 17175674880, 137422176256, 1099444518912, 8795824586752, 70367670435840, 562945658454016, 4503582447501312, 36028728299487232

Offset: 0

N. J. A. Sloane

Numbers whose binary representation is the concatenation of n digits 1 and 2(n-1) digits 0, for n>0. (See A147816.) - Omar E. Pol, Nov 13 2008
a(n) is the number of lattices L in Z^n such that the quotient group Z^n / L is C_8. - Álvar Ibeas, Nov 29 2015
a(n) is a maximum number of intercalates in a Latin square of order 2^n (see A092237). - Eduard I. Vatutin, Apr 30 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..140
Eduard I. Vatutin, Example of Latin squares of order 2^n with maximum number of intercalates.
Index entries for linear recurrences with constant coefficients, signature (12,-32).

Second column of triangle A075499.
Cf. A019677, A147538, A147816.
Cf. A065442, A092237.

[4^(n-1)*(2^n-1): n in [0..40]]; // Vincenzo Librandi, Apr 26 2011

Table[4^(n - 1) (2^n - 1), {n, 0, 19}] (* Michael De Vlieger, Nov 30 2015 *)

a(n)=4^(n-1)*(2^n-1) \\ Charles R Greathouse IV, Oct 07 2015

my(x='x+O('x^30)); concat(0, Vec(x/((1-4*x)*(1-8*x)))) \\ Altug Alkan, Dec 04 2015

[lucas_number1(n,12,32) for n in range(0, 20)] # Zerinvary Lajos, Apr 27 2009

From Barry E. Williams, Jan 17 2000: (Start)
a(n) = ((8^(n+1)) - 4^(n+1))/4.
a(n) = 12a(n-1) - 32a(n-2), n>0; a(0)=1. (End)
a(n) = (4^(n-1))*Stirling2(n+1, 2), n>=0, with Stirling2(n, m)=A008277(n, m).
a(n) = -4^(n-1) + 2*8^(n-1).
E.g.f. for a(n+1), n>=0: d^2/dx^2((((exp(4*x)-1)/4)^2)/2!) = -exp(4*x) + 2*exp(8*x).
G.f.: x/((1-4*x)*(1-8*x)).
((6+sqrt4)^n - (6-sqrt4)^n)/4 in Fibonacci form. Offset 1. a(3)=112. - Al Hakanson (hawkuu(AT)gmail.com), Dec 31 2008
a(n) + A160873(n) + A006096(n) = A006096(n+2), for n > 2. - Álvar Ibeas, Nov 29 2015
Sum_{n>0} 1/a(n) = 4*E - 16/3, where E is the Erdős-Borwein constant (A065442). - Peter McNair, Dec 19 2022

A043569 Numbers whose base-2 representation has exactly 2 runs.

Original entry on oeis.org

2, 4, 6, 8, 12, 14, 16, 24, 28, 30, 32, 48, 56, 60, 62, 64, 96, 112, 120, 124, 126, 128, 192, 224, 240, 248, 252, 254, 256, 384, 448, 480, 496, 504, 508, 510, 512, 768, 896, 960, 992, 1008, 1016, 1020, 1022, 1024, 1536, 1792, 1920, 1984, 2016, 2032, 2040, 2044

Offset: 1

Clark Kimberling

Numbers whose binary representation contains the bit string "10" but not "01". Subsequence of A062289; set difference A062289 minus A101082. - Rick L. Shepherd, Nov 29 2004

Mersenne numbers (A000225) times powers of 2 (A000079). Therefore this sequence contains the even perfect numbers (A000396). - Alonso del Arte, Apr 21 2006

Lei Zhou, Table of n, a(n) for n = 1..10000

Cf. A004736, A007814, A062289, A065442, A101082.

a:=proc(n) local nn,nd: nn:=convert(n,base,2): nd:={seq(nn[j]-nn[j-1],j=2..nops(nn))}: if n=2 then 2 elif nd={0,1} then n else fi end: seq(a(n),n=1..2100); # Emeric Deutsch, Apr 21 2006

Take[Sort[Flatten[Table[(2^x - 1)*(2^y), {x, 32}, {y, 32}]]], 54] (* Alonso del Arte, Apr 21 2006 *)
Select[Range[2500],Length[Split[IntegerDigits[#,2]]]==2&] (* or *) Select[Range[2500],SequenceCount[IntegerDigits[#,2],{1,0}]>0 && SequenceCount[ IntegerDigits[#,2],{0,1}]==0&] (* Harvey P. Dale, Oct 04 2024 *)

def ok(n): b = bin(n)[2:]; return "10" in b and "01" not in b
print([m for m in range(2045) if ok(m)]) # Michael S. Branicky, Feb 04 2021

def a_next(a_n): t = a_n >> 1; return (a_n | t) + (t & 1)
a_n = 2; a = []
for i in range(54): a.append(a_n); a_n = a_next(a_n) # Falk Hüffner, Feb 19 2022

This sequence is twice A023758. - Franklin T. Adams-Watters, Apr 21 2006
Sum_{n>=1} 1/a(n) = A065442. - Amiram Eldar, Feb 20 2022
A007814(a(n)) = A004736(n). - Lorenzo Sauras Altuzarra, Feb 01 2023

More terms from Rick L. Shepherd, Nov 29 2004

A248722 Decimal expansion of Sum_{k>=1} 1/(5^k - 1).

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v, Oct 12 2014

			0.301733853597972457948162159393991192623009431517157720395791923318379825892...

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

Cf. A000005, A065442, A073668, A214369, A248721, A248723, A248724, A248725, A248726.

evalf( add( (1/5)^(n^2)*(1 + 2/(5^n - 1)), n = 1..12), 105); # Peter Bala, Jan 30 2022

x = 1/5; RealDigits[ Sum[ DivisorSigma[0, k] x^k, {k, 1000}], 10, 105][[1]] (* after an observation and the formula of Amarnath Murthy, see A073668 *)

sumpos(k=1,1/(5^k-1)) \\ M. F. Hasler, Oct 15 2014

Equals Sum_{k>=1} d(k)/5^k, where d(k) is the number of divisors of k (A000005). - Amiram Eldar, Jun 22 2020

A248723 Decimal expansion of the Sum_{k>=1} 1/(6^k - 1).

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v, Oct 12 2014

			0.2341491301348092064851116728138729185463610347865138985224213867102381986628...

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

Cf. A000005, A065442, A073668, A214369, A248721, A248722, A248724, A248725, A248726.

evalf(sum(1/(6^k-1), k=1..infinity),120); # Vaclav Kotesovec, Oct 18 2014
# second program with faster converging series
evalf( add( (1/6)^(n^2)*(1 + 2/(6^n - 1)), n = 1..11), 105); # Peter Bala, Jan 30 2022

x = 1/6; 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/(6^k-1)) \\ Michel Marcus, Oct 18 2014

Equals Sum_{k>=1} d(k)/6^k, where d(k) is the number of divisors of k (A000005). - Amiram Eldar, Jun 22 2020

A248724 Decimal expansion of Sum_{k>=1} 1/(7^k - 1).

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v, Oct 12 2014

			0.1909100624102615782021996444176911687692684760082664083347711086409996755846...

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

Cf. A000005, A065442, A073668, A214369, A248721, A248722, A248723, A248725, A248726.

evalf(sum(1/(7^k-1), k=1..infinity),120) # Vaclav Kotesovec, Oct 18 2014
# second program with faster converging series
evalf( add( (1/7)^(n^2)*(1 + 2/(7^n - 1)), n = 1..11), 105); # Peter Bala, Jan 30 2022

x = 1/7; 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/(7^k-1)) \\ Michel Marcus, Oct 18 2014

Equals Sum_{k>=1} d(k)/7^k, where d(k) is the number of divisors of k (A000005). - Amiram Eldar, Jun 22 2020

cp's OEIS Frontend

A214369 Decimal expansion of Sum_{n>=1} 1/(3^n-1).

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A248721 Decimal expansion of Sum_{k>=1} 1/(4^k - 1).

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A065443 Decimal expansion of Sum_{k=1..inf} 1/(2^k-1)^2.

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A173898 Decimal expansion of sum of the reciprocals of the Mersenne primes.

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A240736 Number of compositions of n having exactly one fixed point.

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A016152 a(n) = 4^(n-1)*(2^n-1).

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A043569 Numbers whose base-2 representation has exactly 2 runs.