A020857 -id:A020857 - cp's OEIS frontend

A155693 Decimal expansion of log_2 (22).

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

Offset: 1

N. J. A. Sloane, Oct 30 2009

			4.4594316186372972561993630467257929587032315256817680713128...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for transcendental numbers

Cf. decimal expansion of log_2(m): A020857 (m=3), A020858 (m=5), A020859 (m=6), A020860 (m=7), A020861 (m=9), A020862 (m=10), A020863 (m=11), A020864 (m=12), A152590 (m=13), A154462 (m=14), A154540 (m=15), A154847 (m=17), A154905 (m=18), A154995 (m=19), A155172 (m=20), A155536 (m=21), this sequence, A155793 (m=23), A155921 (m=24).

RealDigits[Log[2, 22], 10, 100][[1]] (* Vincenzo Librandi, Aug 29 2013 *)

Equals 1 + A020863. - Jianing Song, Nov 16 2024

A155793 Decimal expansion of log_2 (23).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 30 2009

			4.5235619560570128722941482441626688444988251254425550594944...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for transcendental numbers

Cf. decimal expansion of log_2(m): A020857 (m=3), A020858 (m=5), A020859 (m=6), A020860 (m=7), A020861 (m=9), A020862 (m=10), A020863 (m=11), A020864 (m=12), A152590 (m=13), A154462 (m=14), A154540 (m=15), A154847 (m=17), A154905 (m=18), A154995 (m=19), A155172 (m=20), A155536 (m=21), A155693 (m=22), this sequence, A155921 (m=24).

RealDigits[Log[2, 23], 10, 100][[1]] (* Vincenzo Librandi, Aug 29 2013 *)

A005664 Denominators of convergents to log_2 3.

Original entry on oeis.org

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

A060528 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the ratios of two tones of musical harmony: the perfect 4th, 4/3 and its complement the perfect 5th, 3/2.

Original entry on oeis.org

1, 2, 3, 5, 7, 12, 29, 41, 53, 200, 253, 306, 359, 665, 8286, 8951, 9616, 10281, 10946, 11611, 12276, 12941, 13606, 14271, 14936, 15601, 31867, 79335, 111202, 190537, 5446238, 5636775, 5827312, 6017849, 6208386, 6398923, 6589460, 6779997, 6970534, 7161071

Offset: 1

Mark William Rankin (MarkRankin95511(AT)Yahoo.com), Apr 12 2001

The sequence was found by a computer search of all the equal divisions of the octave from 1 to over 6589460. This is not a perfect recurrent sequence because its self-accumulating nature fails between the 9th and 10th terms, between the 14th and 15th terms, and between the 30th and 31st terms. The examples of recurrence which are present in this sequence are of the same type that is seen in sequences A054540, A060526 and A060527. The numerical value of each term represents a musical scale based on an equal division of the octave. 12, for example, signifies the scale which is formed by dividing the octave into 12 equal parts. - corrected by K. G. Stier, Jan 29 2015

Also the denominators of increasingly better rational approximations to log(3)/log(2) = 1.5849625... (see A020857). The respective numerators are A254351. The reason why the sequence's "self-accumulating nature fails between the 9th and 10th terms, the 14th and 15th terms and the 30th and 31st terms" (see original comment) is simply that 84/53, 1054/665 and 301994/190537 are very good approximations, thus followed by a jump. (E.g., this phenomenon can also be seen in the numerators and denominators of rational approximations to Pi.). - K. G. Stier, Jan 29 2015

Cf. A054540, A060525, A060526, A060527, A254351.

A005664 is a subsequence, A206788 is a supersequence.

x:bfloat(log(3)/log(2)),fpprec:100, errold:2,for denominator:1 thru 10000 do (numerator:round(x*denominator), errnew:abs(x-numerator/denominator), if errnew < errold then (errold:errnew, print(denominator))); /* K. G. Stier, Jan 29 2015 */

lista(nn) = {d = 2; v = log(3)/log(2); for (den=1, nn, num = round(v*den); newd = abs(v-num/den); if (newd < d, print1(den, ", "); d = newd;););} \\ after Maxima, Michel Marcus, Feb 28 2015

Incorrect term 571611 removed by K. G. Stier, Jan 29 2015

More terms from Jon E. Schoenfield, Feb 06 2015

A098294 a(n) = ceiling(n*log_2(3/2)).

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9, 10, 10, 11, 12, 12, 13, 13, 14, 15, 15, 16, 16, 17, 17, 18, 19, 19, 20, 20, 21, 22, 22, 23, 23, 24, 24, 25, 26, 26, 27, 27, 28, 29, 29, 30, 30, 31, 32, 32, 33, 33, 34, 34, 35, 36, 36, 37, 37, 38, 39, 39, 40, 40, 41, 41, 42, 43, 43, 44

Offset: 0

Wolfdieter Lang, Oct 18 2004

Original name was: Smallest exponent of 2 which gives a power of 2 which is equal to or bigger than (3/2)^n, n = 0,1,... .
Stacking perfect fifths (the frequency ratio of a fifth is 3/2) this sequence determines into which octave the n-th fifth falls. For example, the third fifth, (3/2)^3, falls into the second octave, which means that it lies in the interval [2^1,2^2)=[2,4). The k-th octave comprises ratios in the interval [2^(k-1),2^k), k=1,2,...
Related to the initial number of sequential even terms in an "ideal" sequence under iteration of the 3x+1 Problem on a positive odd value m, where the piecewise function f is given by f(2*m)=m, f(2*m+1)=6*m+4, to ensure f^A122437(n) (m) < m, where n > 1 is the number of odds in the sequence (including m) and floor(1+n*(log(3)/log(2))) is the number of evens. An "ideal" sequence minimizes the effects of f(2*m+1) by following a certain order of even or odd terms along with the rules of the function. A representation of such sequences in terms of parity sequences for values n >= 2 follows:
  n=2, (o,e,e,o,e,e)
  n=3, (o,e,e,o,e,o,e,e)
  n=4, (o,e,e,e,o,e,o,e,o,e,e)
  n=5, (o,e,e,e,o,e,o,e,o,e,o,e,e)
  n=6, (o,e,e,e,e,o,e,o,e,o,e,o,e,o,e,e)
  n=7, (o,e,e,e,e,e,o,e,o,e,o,e,o,e,o,e,o,e,e)
  The pattern is clear, and the formula for the initial number of sequential even terms in each sequence is given by a(n) = floor(1+n*(log(3)/log(2)))-n for n > 1, where the sum of the number of even and odd terms is given by A122437(n) for n > 1. Of course, most values m do not have sequences following this pattern of iteration under f. Also, the reason for placing an extra even term at the end of such sequences is to mitigate to some degree the effects of the possibility that the last odd term is only "slightly" larger than m, i.e., (3*m+1)/4 < m for all m > 1. - Jeffrey R. Goodwin, Aug 25 2011
a(n) gives the position in n-th row of A227048 where (3^n - 2^n) occurs:
A227048(n,a(n)) = A001047(n). - Reinhard Zumkeller, Jun 30 2013
Differs from A005378 at indices n = 0,17,20,22,25,27,29,30,... - M. F. Hasler, Jun 29 2014

			a(0) = 0 because 2^0 = 1 = (3/2)^0 but 2^(-1) = 1/2 < 1.
a(11) = 7 because 2^7 = 128 > 86.497... = (3/2)^11 but 2^6 = 64 < (3/2)^11.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Sonic Studio, Pythagorean Scale.
Eric Weisstein's World of Music, Pythagorean Scale
Wikipedia, Pythagorean tuning
Mike Winkler, On the structure and the behaviour of Collatz 3n + 1 sequences, 2014.

Cf. A098295, A020914, A020857.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a098294 0  = 0
a098294 n  = fromJust (a001047 n `elemIndex` a227048_row n) + 1
-- Reinhard Zumkeller, Jun 30 2013

[0] cat [Floor(1 + n * Log(3)/Log(2)) - n: n in [1..70]]; // Vincenzo Librandi, Jul 13 2015

seq(ceil(n*log[2](3/2)),n=0..100); # Robert Israel, Jul 12 2015

With[{c=Log2[3/2]},Ceiling[c*Range[0,80]]] (* Harvey P. Dale, Feb 24 2024 *)

a(n)=ceil(n*log(3/2)/log(2)) \\ Charles R Greathouse IV, Jul 13 2015

a(n) = !!n + logint(3^n, 2) - n \\ Ruud H.G. van Tol, Nov 21 2023

2^a(n) >= (3/2)^n but 2^(a(n) - 1) < (3/2)^n, n >= 0.
a(n) = ceiling(tau*n) with tau := log(3)/log(2) - 1  = 0.584962501..., n >= 0.
a(n) = floor(1 + n * log(3)/log(2)) - n, n >= 1. - Mike Winkler, Dec 31 2010

A022921 Number of integers m such that 3^n < 2^m < 3^(n+1).

Original entry on oeis.org

1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 2, 2, 1, 2, 1

Offset: 0

Clark Kimberling

Represents increments between successive terms of allowable dropping times in the Collatz (3x+1) problem. That is, a(n) = A020914(n+1) - A020914(n). - K. Spage, Oct 23 2009

			From _Amiram Eldar_, Mar 01 2024: (Start)
a(0) = 1 because 3^0 = 1 < 2^1 = 2 < 3^1 = 3.
a(1) = 2 because 3^1 = 3 < 2^2 = 4 < 2^3 = 8 < 3^2 = 9.
a(2) = 1 because 3^2 = 9 < 2^4 = 16 < 3^3 = 27. (End)

T. D. Noe, Table of n, a(n) for n = 0..1000
Mike Winkler, The algorithmic structure of the finite stopping time behavior of the 3x + 1 function, arXiv:1709.03385 [math.GM], 2017.

Cf. A020914, A056576, A076227, A100982, A122437.

See also A020857 (decimal expansion of log_2(3)).

Digits := 100: c1 := log(3.)/log(2.): A022921 := n->floor((n+1)*c1)-floor(n*c1);
seq(ilog2(3^(n+1)) - ilog2(3^n), n=0 .. 1000); # Robert Israel, Dec 11 2014

i2 = 1; Table[p = i2; While[i2++; 2^i2 < 3^(n + 1)]; i2 - p, {n, 0, 98}] (* T. D. Noe, Feb 28 2014 *)
f[n_] := Floor[ Log2[ 3^n] + 1]; Differences@ Array[f, 106, 0] (* Robert G. Wilson v, May 25 2014 *)

a(n) = logint(3^(n+1),2) - logint(3^n,2) \\ Ruud H.G. van Tol, Dec 28 2022

Vec(matreduce([logint(2^i,3)|i<-[1..158]])[,2])[1..-2] \\ Ruud H.G. van Tol, Dec 29 2022

a(n) = floor((n+1)*log_2(3)) - floor(n*log_2(3)).
a(n) = A122437(n+2) - A122437(n+1) - 1. - K. Spage, Oct 23 2009
First differences of A020914. - Robert G. Wilson v, May 25 2014
First differences of A056576. - L. Edson Jeffery, Dec 12 2014
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = log_2(3) (A020857). - Amiram Eldar, Mar 01 2024

A234515 Natural numbers n sorted by decreasing values of number k(n) = log_n(sigma(n)), where sigma(n) = A000203(n) = the sum of divisors of n.

Original entry on oeis.org

2, 4, 6, 12, 8, 24, 18, 3, 36, 30, 10, 60, 20, 48, 16, 72, 120, 84, 42, 40, 180, 90, 96, 28, 144, 240, 168, 14, 108, 360, 54, 32, 420, 80, 252, 132, 216, 56, 210, 126, 300, 66, 336, 480, 192, 288, 720, 840, 156, 504, 150, 540, 264, 140, 600, 78, 270, 1260, 432

Offset: 1

Jaroslav Krizek, Jan 03 2014

nonn

Number k(n) = log_n(sigma(n)) = log(sigma(n)) / log(n) is number such that n^k(n) = sigma(n).
The last term of this infinite sequence is number 1, k(1) = 1 (minimal value of function k(n)).
Conjecture: Every natural number n has a unique value of number k(n).
See A234517 - sequence of numbers a(n) such that a(n) > a(k) for all k < n.

			For number 2; k(2) = log_2(sigma(2)) = log_2(3) = 1,5849625007… = A020857 (maximal value of function k(n)).

Jaroslav Krizek, Table of n, a(n) for n = 1..200

Cf. A234516, A234517, A234518, A234519, A234520, A234521, A234522, A234523, A234524.

lista(nn=100000) = {v = vector(nn, n, if (n==1, 0, log(sigma(n))/log(n))); v = vecsort(v,,5); for (i=1, 80, print1(v[i], ", "));} \\ Michel Marcus, Dec 11 2014

A236020 Natural numbers n sorted by increasing values of k(n) = log_tau(n) (sigma(n)), where sigma(n) = A000203(n) = the sum of divisors of n and tau(n) = A000005(n) = the number of divisors of n.

Original entry on oeis.org

1, 2, 4, 6, 12, 8, 24, 3, 18, 36, 30, 60, 10, 20, 48, 72, 120, 16, 40, 84, 180, 42, 90, 240, 144, 360, 96, 168, 28, 420, 108, 80, 252, 720, 14, 15, 210, 840, 54, 56, 336, 480, 216, 126, 32, 504, 288, 9, 540, 1260, 300, 132, 140, 1680, 192, 2520, 1080, 600, 630

Offset: 1

Jaroslav Krizek, Jan 18 2014

nonn

The number k(n) = log_tau(n) (sigma(n)) = log(sigma(n)) / log(tau(n)) is such that tau(n)^k(n) = sigma(n).
Conjecture: every natural number n has a unique value of k(n). [The conjecture is wrong: e.g., k(5) = k(22) = log(6)/log(2). - Amiram Eldar, Jan 17 2021]
See A236021 - sequence of numbers a(n) such that a(n) > a(k) for all k < n.

			For number 1; k(1) = 1.
For number 2; k(2) = log_tau(2) (sigma(2)) = log_2 (3) = 1.5849625007... = A020857.

Michel Marcus, Table of n, a(n) for n = 1..1288

Cf. A000005, A000203, A236021.

Cf. A236022, A236023, A236024, A236025, A236026.

A[nn_] := Ordering[ N[ Join[ {1}, Table[ Log[DivisorSigma[0, i], DivisorSigma[1, i]], {i, 2, nn} ] ] ] ];
A236020[nn_] := A[nn^2][[1 ;; nn]];
A236020[59] (* Robert P. P. McKone, Jan 17 2021 *)

\\ warning: does not generate all the terms up to nn
f(k) = if (k==1, 1, log(sigma(k)) / log(numdiv(k)));
lista(nn) = vecsort(vector(nn, k, f(k)),, 1); \\ Michel Marcus, Jan 16 2021

A191475 Values of i in the numbers 2^i*3^j, i >= 1, j >= 1 (A033845).

Original entry on oeis.org

1, 2, 1, 3, 2, 4, 1, 3, 5, 2, 4, 1, 6, 3, 5, 2, 7, 4, 1, 6, 3, 8, 5, 2, 7, 4, 1, 9, 6, 3, 8, 5, 2, 10, 7, 4, 1, 9, 6, 3, 11, 8, 5, 2, 10, 7, 4, 12, 1, 9, 6, 3, 11, 8, 5, 13, 2, 10, 7, 4, 12, 1, 9, 6, 14, 3, 11, 8, 5, 13, 2, 10, 7, 15, 4, 12, 1, 9, 6, 14, 3, 11

Offset: 1

Zak Seidov, Aug 30 2012

nonn

Signature sequence of log_2(3) (A020857). - R. J. Mathar, May 27 2024

			a(10) = 2 because A033845(10) = 108 = 2^2*3^3.
a(100) = 2 because A033845(100) = 59872 = 2^8*3^7.
a(1000) = 56 because A033845(1000) = 216172782113783808 = 2^56*3^1.

Cf. A003586 (numbers 2^i*3^j, i >= 0, j >= 0), A033845 (numbers 2^i*3^j, i >= 1, j >= 1), A191476 (values of j), A020857.

mx = 1000000; t = Select[Sort[Flatten[Table[2^i 3^j, {i, Log[2, mx]}, {j, Log[3, mx]}]]], # <= mx &]; Table[FactorInteger[i][[1, 2]], {i, t}] (* T. D. Noe, Aug 31 2012 *)

from sympy import integer_log
def A191475(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum((x//3**i).bit_length() for i in range(integer_log(x,3)[0]+1))
    return 1+(~(m:=bisection(f,n,n))&m-1).bit_length() # Chai Wah Wu, Sep 15 2024

Edited by N. J. A. Sloane, May 26 2024

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

cp's OEIS Frontend

A155693 Decimal expansion of log_2 (22).

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A155793 Decimal expansion of log_2 (23).

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Mathematica

A005664 Denominators of convergents to log_2 3.

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A060528 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the ratios of two tones of musical harmony: the perfect 4th, 4/3 and its complement the perfect 5th, 3/2.

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Maxima

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A098294 a(n) = ceiling(n*log_2(3/2)).

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Magma

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A022921 Number of integers m such that 3^n < 2^m < 3^(n+1).

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Maple

Mathematica

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A234515 Natural numbers n sorted by decreasing values of number k(n) = log_n(sigma(n)), where sigma(n) = A000203(n) = the sum of divisors of n.

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