A084255 -id:A084255 - cp's OEIS frontend

A076576 Erroneous version of A084255.

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

Offset: 0

dead

A064442 Decimal expansion of number with continued fraction expansion 2, 3, 5, 7, 11, 13, 17, 19, ... = 2.3130367364335829063839516 ...

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

Offset: 1

Robert G. Wilson v, Oct 01 2001

Continued fraction expansion of the prime numbers. - Harvey P. Dale, Sep 25 2012

			2.313036736433582906383951602641782476396689771803256340210124442144564731776...

MathOverflow, What is the effect of adding 1/2 to a continued fraction?

Cf. A060997, A302937.

RealDigits[ N[ FromContinuedFraction[ Table[ Prime[n], {n, 1, 100} ]], 100]] [[1]]
RealDigits[FromContinuedFraction[Prime[Range[200]]],10,120][[1]] (* Harvey P. Dale, Sep 06 2021 *)

1/A084255. - Franklin T. Adams-Watters, Jul 31 2009
From Peter Bala, Nov 26 2019: (Start)
Denoting the constant by c we have the related simple continued fraction expansions (prime(n) denotes the n-th prime number):
2*c = [4; 1, 1, 1, 2, 14, 5, 1, 1, 6, 34, 9, 1, 1, 11, 58, 15, 1, 1, 18, 82, 21, ..., 1, 1, (prime(3*n) - 1)/2, 2*prime(3*n+1), (prime(3*n+2) - 1)/2, ...];
(1/2)*c = [1; 6, 2, 1, 1, 3, 22, 6, 1, 1, 8, 38, 11, 1, 1, 14, 62, 18, 1, 1, 20, 86, 23, ..., 1, 1, (prime(3*n+1) - 1)/2, 2*prime(3*n+2), (prime(3*n+3) - 1)/2, ...];
(c + 1)/(c - 1) = [2; 1, 1, 10, 3, 1, 1, 5, 26, 8, 1, 1, 9, 46, 14, 1, 1, 15, 74, 20, ..., 1, 1, (prime(3*n+2) - 1)/2, 2*prime(3*n+3), (prime(3*n+4) - 1)/2, ...]. (End)

A084256 Decimal expansion of x such that x^2 + x^3 + x^5 + x^7 + x^11 + x^13 + x^17 + ... = 1.

Original entry on oeis.org

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

Offset: 0

Frank Ellermann, May 23 2003

A constant appearing in the asymptotic formula for the number of prime additive compositions. See Finch's article.

			0.67740177613066042797630631643196719199252141284...

Steven R. Finch, Kalmar's composition constant, June 5, 2003. [Cached copy, with permission of the author]
G. L. Honaker, Jr., Prime Curios!: 0.6774017761306604279...
Prime Curios, Michael Hartley.

Cf. A000040, A084255, A084257.

RealDigits[x/.FindRoot[1==Sum[x^Prime[n], {n, 150}], {x, {0, 1}}, WorkingPrecision->100]][[1]] (* T. D. Noe, Oct 14 2003 *)

More terms from T. D. Noe, Oct 14 2003

A152062 Decimal expansion of number with continued fraction expansion 1, 2, 3, 5, 7, 11, 13, 17, 19, ...

Original entry on oeis.org

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

Offset: 1

Frank S. Thomas (fthomas(AT)physik.uni-wuerzburg.de), Nov 22 2008

			1.4323320871859028689092537...

Cf. A064442, A127549.

RealDigits[N[FromContinuedFraction[Prepend[Table[Prime[n], {n, 1, 100}], 1]], 100]][[1]]

One plus A084255. [R. J. Mathar, Nov 27 2008]

Equals 1+1/(2+1/(3+1/(5+1/(7+1/(11+...))))). - Daniel Forgues, Mar 08 2016

A247847 Decimal expansion of m = (1-1/e^2)/2, one of Renyi's parking constants.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Sep 25 2014

Curiously, this Renyi parking constant is very close to the prime generated continued fraction A084255 (gap ~ 10^-7).

			0.432332358381693654053000252513757798296184227045212...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.3 Renyi's parking constant, p. 280.

Eric Weisstein's MathWorld, Rényi's Parking Constants
Marek Wolf, Continued fractions constructed from prime numbers, arxiv.org/abs/1003.4015, pp. 4-5.

Cf. A001113, A050996, A073333, A084255, A092555, A242943, A243266, A247392.

RealDigits[(1 - 1/E^2)/2 , 10, 104] // First

Define s(n) = Sum_{k = 0..n} 2^k/k!. Then (1 - 1/e^2)/2 = Sum_{n >= 0} 2^n/( (n+1)!*s(n)*s(n+1) ). Cf. A073333. - Peter Bala, Oct 23 2023

A302937 Decimal expansion of continued fraction 1/(4+1/(6+1/(8+1/(9+1/(10+...))))).

Original entry on oeis.org

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

Offset: 0

Paolo P. Lava, Apr 16 2018

Decimal expansion of the constant whose continued fraction form is the sequence of all the composite numbers.

			0.24019347271354040290765990165361943880486990402195476057141424611256753...

Cf. A002808, A084255.

with(numtheory); P:=proc(q) local a,n; a:=0; for n from q by -1 to 4 do if not isprime(n) then a:=1/(a+n); fi; od; print(evalf(a,120)); end: P(10^4);

RealDigits[ Fold[1/(#1 + #2) &, 1, Reverse[ Composite@# & /@ Range@40]], 10, 111][[1]] (* Robert G. Wilson v, Apr 29 2018 *)

A191608 Decimal expansion of number whose continued fraction is based on noncomposite numbers.

Original entry on oeis.org

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

Offset: 0

Fabrice Auzanneau, Jun 08 2011

			0.6981621154383939035335... = 1/(1+1/(2+1/(3+1/(5+1/(7+1/(11+1/(13+1/...))))))).

Equals 1/(1+A084255). - Nathaniel Johnston, Jun 08 2011

Equals 1/A152062. - R. J. Mathar, Jun 17 2011

Corrected and extended by Nathaniel Johnston, Jun 08 2011

A328726 Decimal expansion of the number with continued fraction expansion 4, 6, 8, 9, 10, 12, 14, 15, ... (A002808 = composite numbers).

Original entry on oeis.org

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

Offset: 1

Alexander Victorius Budianto, Oct 26 2019

			4.163310470941149346202768593813039507043958...

Cf. A002808, A060997, A064442, A084255, A127551, A302937.

Equals 1/A302937. - Alois P. Heinz, Nov 13 2019

More digits from Alois P. Heinz, Nov 13 2019

A330867 Decimal expansion of the continued fraction 1/(1 + 2/(2 + 3/(3 + 5/(5 + 7/(7 + ... + prime(k)/(prime(k) + ...)))))).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Apr 28 2020

			0.58152500459221465439915170481800446195586754...

Eric Weisstein's World of Mathematics, Generalized Continued Fraction

Cf. A000040, A064442, A068996, A073333, A084255, A085825, A152062, A191504, A191815, A233588, A247856.

cp's OEIS Frontend

A076576 Erroneous version of A084255.

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A064442 Decimal expansion of number with continued fraction expansion 2, 3, 5, 7, 11, 13, 17, 19, ... = 2.3130367364335829063839516 ...

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A084256 Decimal expansion of x such that x^2 + x^3 + x^5 + x^7 + x^11 + x^13 + x^17 + ... = 1.

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A152062 Decimal expansion of number with continued fraction expansion 1, 2, 3, 5, 7, 11, 13, 17, 19, ...

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A247847 Decimal expansion of m = (1-1/e^2)/2, one of Renyi's parking constants.

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A302937 Decimal expansion of continued fraction 1/(4+1/(6+1/(8+1/(9+1/(10+...))))).

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A191608 Decimal expansion of number whose continued fraction is based on noncomposite numbers.

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A328726 Decimal expansion of the number with continued fraction expansion 4, 6, 8, 9, 10, 12, 14, 15, ... (A002808 = composite numbers).

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A330867 Decimal expansion of the continued fraction 1/(1 + 2/(2 + 3/(3 + 5/(5 + 7/(7 + ... + prime(k)/(prime(k) + ...)))))).

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