A028507 -id:A028507 - 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

A005663 Numerators of convergents to log_2(3) = log(3)/log(2).

Original entry on oeis.org

1, 2, 3, 8, 19, 65, 84, 485, 1054, 24727, 50508, 125743, 176251, 301994, 16785921, 17087915, 85137581, 272500658, 357638239, 630138897, 9809721694, 10439860591, 103768467013, 217976794617, 1193652440098, 8573543875303

Offset: 0

N. J. A. Sloane

			log_2(3) = 1.5849625007211561814537389439...

R. K. Guy, personal communication.
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
R. E. Crandall, On the 3x+1 problem, Math. Comp., 32 (1978) 1281-1292.
E. G. Dunne, Pianos and Continued Fractions
E. G. Dunne, Pianos and Continued Fractions
R. K. Guy, Letter to N. J. A. Sloane, 1977
Eric Weisstein's World of Music, Comma of Pythagoras

Cf. A005664, A028507, A020857.

Numerator[Convergents[Log[2,3],30]] (* Harvey P. Dale, Sep 10 2015 *)

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

More terms from James Sellers, Sep 16 2000

A206788 Denominators of semiconvergents to log_2(3), which equals log(3)/log(2).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 12, 17, 29, 41, 53, 94, 147, 200, 253, 306, 359, 665, 971, 1636, 2301, 2966, 3631, 4296, 4961, 5626, 6291, 6956, 7621, 8286, 8951, 9616, 10281, 10946, 11611, 12276, 12941, 13606, 14271, 14936, 15601, 16266, 31867, 47468, 79335, 111202, 190537

Offset: 1

Keenan Pepper, Feb 12 2012

These are also equal divisions of the octave that have good approximations of the 3rd harmonic (and hence other Pythagorean intervals such as 3/2, 4/3, 9/8 etc.).

The semiconvergents are important musically as the cardinalities of MOS scales of Pythagorean tuning.

Wikipedia, Definition of continued fraction semiconvergents

Cf. A028507 (continued fraction expansion), A005664 (denominators of convergents).

cp's OEIS Frontend

A005664 Denominators of convergents to log_2 3.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A005663 Numerators of convergents to log_2(3) = log(3)/log(2).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A206788 Denominators of semiconvergents to log_2(3), which equals log(3)/log(2).

Views

Author

Keywords

Comments

Links

Crossrefs