A005663 -id:A005663 - cp's OEIS frontend

A005664 Denominators of convergents to log_2 3.

1, 1, 2, 5, 12, 41, 53, 306, 665, 15601, 31867, 79335, 111202, 190537, 10590737, 10781274, 53715833, 171928773, 225644606, 397573379, 6189245291, 6586818670, 65470613321, 137528045312, 753110839881, 5409303924479, 6162414764360

Offset: 0

N. J. A. Sloane, R. K. Guy

			log_2 3 = 1.5849625007211561814537389439...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..200
Oliver K. Clay, The Long Search for Collatz Counterexamples, J. Humanistic Math. (2023) Vol. 13, No. 2, 199-227. See p. 205.
R. E. Crandall, On the 3x+1 problem, Math. Comp., 32 (1978) 1281-1292.
E. G. Dunne, Pianos and Continued Fractions
R. K. Guy, Letter to N. J. A. Sloane, 1977
David Ryan, An algorithm to assign musical prime commas to every prime number and construct a universal and compact free Just Intonation musical notation, Preprint, arXiv preprint arXiv:1612.01860 [cs.SD], 2016.
Eric Weisstein's World of Music, Comma of Pythagoras

Cf. A005663, A028507, A020857.

Convergents[Log[2, 3], 30] // Denominator (* Jean-François Alcover, Dec 12 2016 *)

a(n) = component(component(contfracpnqn(contfrac(log(3)/log(2), n)), 1), 2) \\ Michel Marcus, May 20 2013

More terms from James Sellers, Sep 16 2000

A221363 Decimal expansion of the Pythagorean comma.

Original entry on oeis.org

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

Offset: 1

Jonathan Sondow, Jan 19 2013

In musical tuning, the Pythagorean comma is 12 fifths / 7 octaves = (3/2)^12 / 2^7.

			3^12 / 2^19 = 531441/524288 = 1.0136432647705078125

Larry Baggett, In the Dark on the Sunny Side: A Memoir of an Out-of-Sight Mathematician, Mathematical Association of America, 2012, p. 78.
Dave Benson, Music: A Mathematical Offering. Cambridge: Cambridge University Press (2006): 164.
J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, 1995, p. 257.

Peter A. Frazer, Temperament, Midicode, January 2010.
Eric Weisstein's World of Music, Comma of Pythagoras
Wikipedia, Pythagorean comma
OEIS index entry for music
The multi-faceted reach of the OEIS: Music

Cf. A005663, A005664, A010774, A046102.

Mathematica
```
RealDigits[N[31441/524288, 50]][[1]]
```

A229948/A229943 - Omar E. Pol, Oct 25 2013

A028507 Continued fraction expansion for log_2(3).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 1, 5, 2, 23, 2, 2, 1, 1, 55, 1, 4, 3, 1, 1, 15, 1, 9, 2, 5, 7, 1, 1, 4, 8, 1, 11, 1, 20, 2, 1, 10, 1, 4, 1, 1, 1, 1, 1, 37, 4, 55, 1, 1, 49, 1, 1, 1, 4, 1, 3, 2, 3, 3, 1, 5, 16, 2, 3, 1, 1, 1, 1, 1, 5, 2, 1, 2, 8, 7, 1, 1, 2, 1, 1, 3, 3, 1, 1, 1, 1, 5, 4, 2, 2, 2, 16, 8, 10, 1, 25, 2, 1

Offset: 0

Tony Smith (tsmith(AT)innerx.net)

			log_2(3) = 1.5849625007211561814537389439...

T. D. Noe, Table of n, a(n) for n = 0..9999
E. G. Dunne, Pianos and Continued Fractions
Terence Jackson and Keith Matthews, "On Shanks' Algorithm for Computing the Continued Fraction of log_b a" , Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.7
T. H. Jackson & K. R. Matthews, The 1000 partial quotients of log_2(3)
Dave Rusin, Why 12 tones per octave? [Broken link]
Dave Rusin, Why 12 tones per octave? [Cached copy]

Cf. A005663, A005664, A020857 (decimal expansion).

Digits := 200: convert(evalf( log(3)/log(2) ),confrac);

ContinuedFraction[Log[2,3],120] (* Harvey P. Dale, Oct 24 2011 *)

More terms from James Sellers, Sep 16 2000

Offset changed by Andrew Howroyd, Aug 07 2024

A042937 Denominators of continued fraction convergents to sqrt(1000).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 53, 114, 281, 4329, 8939, 22207, 142181, 164388, 306569, 470957, 777526, 1248483, 78183472, 79431955, 157615427, 237047382, 394662809, 631710191, 4184923955, 9001558101, 22188040157, 341822160456, 705832361069, 1753486882594

Offset: 0

N. J. A. Sloane

			sqrt(1000) = 31.62... = 31 + 1/(1 + 1/(1 + ...)) with convergents 31/1, 32/1, 63/2, 95/3, 158/5, ... - _M. F. Hasler_, Nov 02 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 78960998, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1).

Cf. A042936 (numerators), A040968 (continued fraction), A010467 (decimals).

Analog for sqrt(m): A000129 (m=2), A002530 (m=3), A001076 (m=5), A041007 (m=6), A041009 (m=7), A041011 (m=8), A005663 (m=10), A041015 (m=11), A041017 (m=12), ..., A042933 (m=998), A042935 (m=999).

Denominator[Convergents[Sqrt[1000], 30]] (* Vincenzo Librandi, Feb 01 2014 *)

A42937=contfracpnqn(c=contfrac(sqrt(1000)),#c-1)[2,] \\ Possibly incorrect last term ignored. NB: a(n) = A42937[n+1]. For more terms use e.g. \p999, or compute any a(n) from this as in A042936. - M. F. Hasler, Nov 01 2019

More terms from Vincenzo Librandi, Feb 01 2014

A355512 Sum of numerator and denominator in the convergents of the approximation of log(2)/log(3) by a continued fraction.

Original entry on oeis.org

2, 3, 5, 13, 31, 106, 137, 791, 1719, 40328, 82375, 205078, 287453, 492531, 27376658, 27869189, 138853414, 444429431, 583282845, 1027712276, 15998966985, 17026679261, 169239080334, 355504839929, 1946763279979, 13982847799782, 15929611079761, 29912458879543, 135579446597933

Offset: 1

Hugo Pfoertner, Jul 05 2022

nonn

Cf. A005663, A005664, A102525, A355513, A355515.

Cf. A355514 for the relation to potential cycle lengths of the 3x+1 problem.

a355512(upto) = {my(q=log(2)/log(3), fa=oo); for (denmax=1, upto, my(f=bestappr(q,denmax)); if (fa!=f, print1(numerator(f)+denominator(f),", "); fa=f))};
\\ needs increased precision for larger terms
a355512(10^6)

\\ See also A005663 and A005664 for efficient code.

A046102 Denominators of convergents to the comma of Pythagoras.

Original entry on oeis.org

1, 1, 3, 7, 24, 31, 179, 389, 9126, 18641, 46408, 65049, 111457, 6195184, 6306641, 31421748, 100571885, 131993633, 232565518, 3620476403, 3853041921, 38297853692, 80448749305, 440541600217, 3164239950824, 3604781551041

Offset: 1

Eric W. Weisstein

T. D. Noe, Table of n, a(n) for n=1..200
Eric Weisstein's World of Music, Comma of Pythagoras

Cf. A005664.

Denominator[ Table[ ContinuedFraction[ Log[ 2 ]/Log[ 3/2 ], i ]//Normal, {i, 30} ] ]

a(n) = A005663(n) - A005664(n). - T. D. Noe, Oct 12 2007

cp's OEIS Frontend

A005664 Denominators of convergents to log_2 3.

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A221363 Decimal expansion of the Pythagorean comma.

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A028507 Continued fraction expansion for log_2(3).

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Maple

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A042937 Denominators of continued fraction convergents to sqrt(1000).

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Mathematica

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A355512 Sum of numerator and denominator in the convergents of the approximation of log(2)/log(3) by a continued fraction.

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PARI

PARI

A046102 Denominators of convergents to the comma of Pythagoras.

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Mathematica

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