A016629 -id:A016629 - cp's OEIS frontend

A152683 Decimal expansion of log_6 (2).

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

Offset: 0

N. J. A. Sloane, Oct 30 2009

The upper bound for the ratio of the number of 3x+1 steps to all steps in the Collatz iteration. - T. D. Noe, Apr 30 2010

			.38685280723454158687024613846782087646514185945710342838949...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Emmanuel Jeandel, Michael Rao, An aperiodic set of 11 Wang tiles, arXiv:1506.06492 [cs.DM], 2015. See p. 10.
Index entries for transcendental numbers

Cf. decimal expansion of log_6(m): this sequence, A152935 (m=3), A153102 (m=4), A153202 (m=5), A153617 (m=7), A153754 (m=8), A154009 (m=9), A154157 (m=10), A154178 (m=11), A154199 (m=12), A154278 (m=13), A154466 (m=14), A154567 (m=15), A154776 (m=16), A154856 (m=17), A154911 (m=18), A155044 (m=19), A155490 (m=20), A155554 (m=21), A155697 (m=22), A155823 (m=23), A155959 (m=24).

SetDefaultRealField(RealField(100)); Log(2)/Log(6); // G. C. Greubel, Sep 13 2018

RealDigits[Log[6,2],10,120][[1]] (* Harvey P. Dale, Sep 12 2012 *)

default(realprecision, 100); log(2)/log(6) \\ G. C. Greubel, Sep 13 2018

Equals log(2)/log(6) (A002162/A016629), that is, log(2)/(log(2)+log(3)). - Michel Marcus, Aug 18 2018

A154009 Decimal expansion of log_6 (9).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 30 2009

			1.2262943855309168262595077230643582470697162810857931432210...

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

Cf. decimal expansion of log_6(m): A152683 (m=2), A152935 (m=3), A153102 (m=4), A153202 (m=5), A153617 (m=7), A153754 (m=8), this sequence, A154157 (m=10), A154178 (m=11), A154199 (m=12), A154278 (m=13), A154466 (m=14), A154567 (m=15), A154776 (m=16), A154856 (m=17), A154911 (m=18), A155044 (m=19), A155490 (m=20), A155554 (m=21), A155697 (m=22), A155823 (m=23), A155959 (m=24).

SetDefaultRealField(RealField(100)); Log(9)/Log(6); // G. C. Greubel, Sep 14 2018

RealDigits[Log[6, 9], 10, 100][[1]] (* Vincenzo Librandi, Aug 31 2013 *)

default(realprecision, 100); log(9)/log(6) \\ G. C. Greubel, Sep 14 2018

Equals A016632 / A016629 =2/(1+A102525). - R. J. Mathar, Jul 29 2024

A279060 Number of divisors of n of the form 6*k + 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 3, 2, 1, 2, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 4, 1, 2, 1, 2, 1, 2, 3, 1, 2, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 1, 2, 2, 1, 2, 1, 2, 1

Offset: 0

Ilya Gutkovskiy, Dec 05 2016

Möbius transform is the period-6 sequence {1, 0, 0, 0, 0, 0, ...}.

			a(14) = 2 because 14 has 4 divisors {1,2,7,14} among which 2 divisors {1,7} are of the form 6*k + 1.

Antti Karttunen, Table of n, a(n) for n = 0..65537
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A001227, A001817, A001826, A001876, A035218, A140213, A188169, A319995, A320001.

Cf. A001620, A016629, A222457 (psi(1/6)).

nmax = 120; CoefficientList[Series[Sum[x^k/(1 - x^(6 k)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 120; CoefficientList[Series[Sum[x^(6 k + 1)/(1 - x^(6 k + 1)), {k, 0, nmax}], {x, 0, nmax}], x]
Table[Count[Divisors[n],?(Mod[#,6]==1&)],{n,0,120}] (* _Harvey P. Dale, Apr 27 2018 *)

A279060(n) = if(!n,n,sumdiv(n, d, (1==(d%6)))); \\ Antti Karttunen, Jul 09 2017

from sympy import divisors
def A279060(n): return sum(d%6 == 1 for d in divisors(n)) # David Radcliffe, Jun 19 2025

G.f.: Sum_{k>=1} x^k/(1 - x^(6*k)).
G.f.: Sum_{k>=0} x^(6*k+1)/(1 - x^(6*k+1)).
From Antti Karttunen, Oct 03 2018: (Start)
a(n) = A320001(n) + [1 == n (mod 6)], where [ ] is the Iverson bracket, giving 1 only when n = 1 mod 6, and 0 otherwise.
a(n) = A035218(n) - A319995(n). (End)
a(n) = (A035218(n) + A035178(n)) / 2. - David Radcliffe, Jun 19 2025
Sum_{k=1..n} a(k) = n*log(n)/6 + c*n + O(n^(1/3)*log(n)), where c = gamma(1,6) - (1 - gamma)/6 = 0.686263..., gamma(1,6) = -(psi(1/6) + log(6))/6 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A319995 Number of divisors of n of the form 6*k + 5.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 2, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 2, 0, 0, 1, 1, 1, 0, 0, 0, 0, 2, 1, 0, 1, 1, 2, 1, 0, 0, 0, 1, 0, 2, 0, 0, 1, 0, 1, 1, 0, 2, 0, 1, 1, 1, 1, 0, 1, 0, 1, 2, 0, 0, 0, 1, 1, 1, 1, 0, 0, 2

Offset: 1

Antti Karttunen, Oct 03 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A035218, A279060, A320005.

Cf. A001620, A016629, A222458 (psi(5/6)).

a[n_] := DivisorSum[n, 1 &, Mod[#, 6] == 5 &]; Array[a, 100] (* Amiram Eldar, Nov 25 2023 *)

A319995(n) = if(!n,n,sumdiv(n, d, (5==(d%6))));

a(n) = A035218(n) - A279060(n).
G.f.: Sum_{k>=1} x^(5*k)/(1 - x^(6*k)). - Ilya Gutkovskiy, Sep 11 2019
Sum_{k=1..n} a(k) = n*log(n)/6 + c*n + O(n^(1/3)*log(n)), where c = gamma(5,6) - (1 - gamma)/6 = -0.220635..., gamma(5,6) = -(psi(5/6) + log(6))/6 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A320001 Number of proper divisors of n of the form 6*k + 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 3, 1, 2, 1, 1, 1, 2, 2

Offset: 1

Antti Karttunen, Oct 03 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A279060, A293895, A319991, A320003, A320005, A320015.

Cf. A001620, A016629, A222457 (psi(1/6)).

a[n_] := DivisorSum[n, 1 &, # < n && Mod[#, 6] == 1 &]; Array[a, 100] (* Amiram Eldar, Nov 25 2023 *)

A320001(n) = if(!n,n,sumdiv(n, d, (d

a(n) = A279060(n) - [+1 = n (mod 6)], where [ ] is the Iverson bracket, giving 1 only when n = 1 mod 6, and 0 otherwise.
a(n) = A320015(n) - A320005(n).
a(n) = A007814(A319991(n)).
G.f.: Sum_{k>=1} x^(12*k-10) / (1 - x^(6*k-5)). - Ilya Gutkovskiy, Apr 14 2021
Sum_{k=1..n} a(k) = n*log(n)/6 + c*n + O(n^(1/3)*log(n)), where c = gamma(1,6) - (2 - gamma)/6 = 0.519597..., gamma(1,6) = -(psi(1/6) + log(6))/6 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A320005 Number of proper divisors of n of the form 6*k + 5.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 2, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 2, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 2, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 2

Offset: 1

Antti Karttunen, Oct 03 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A293896, A319992, A319995, A320001, A320003, A320015.

Cf. A001620, A016629, A222458 (psi(5/6)).

a[n_] := DivisorSum[n, 1 &, # < n && Mod[#, 6] == 5 &]; Array[a, 100] (* Amiram Eldar, Nov 25 2023 *)

A320005(n) = if(!n,n,sumdiv(n, d, (d

a(n) = A319995(n) - [+5 = n (mod 6)], where [ ] is the Iverson bracket, giving 1 only when n = -1 mod 6, and 0 otherwise.
a(n) = A320015(n) - A320001(n).
a(n) = A007814(A319992(n)).
G.f.: Sum_{k>=1} x^(12*k-2) / (1 - x^(6*k-1)). - Ilya Gutkovskiy, Apr 14 2021
Sum_{k=1..n} a(k) = n*log(n)/6 + c*n + O(n^(1/3)*log(n)), where c = gamma(5,6) - (2 - gamma)/6 = -0.387302..., gamma(5,6) = -(psi(5/6) + log(6))/6 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A173160 Decimal expansion of the constant x satisfying x^x = 6.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 11 2010

			2.2318286..^2.2318286..=6. 2.2318286..*log(2.2318286..) = A016629.

Cf. A030437, A030797, A030798, A173158, A173159.

x=6;RealDigits[Log[x]/ProductLog[Log[x]],10,6! ][[1]]

Digits of log(6)/W(log(6)).

Keyword:cons added by R. J. Mathar, Mar 14 2010

A320003 Number of proper divisors of n of the form 6*k + 3.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 2, 0, 0, 3, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 0, 1, 0, 0, 2, 0, 0, 3, 0, 0, 2, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 3, 0, 0, 2, 0, 0, 1, 0, 0, 4, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 0, 2, 0, 0, 3

Offset: 1

Antti Karttunen, Oct 03 2018

Number of divisors of n that are odd multiples of 3 and less than n.

			For n = 18, of its five proper divisors [1, 2, 3, 6, 9] only 3 and 9 are odd multiples of three, thus a(18) = 2.
For n = 108, the odd part is 27 for which 27/3 has 3 divisors. As 108 is even, we don't subtract 1 from that 3 to get a(108) = 3. - _David A. Corneth_, Oct 03 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A319990, A320001, A320005.

Cf. A001620, A016629, A020759 (psi(1/2)).

a[n_] := DivisorSum[n, 1 &, # < n && Mod[#, 6] == 3 &]; Array[a, 100] (* Amiram Eldar, Nov 25 2023 *)

A320003(n) = if(!n,n,sumdiv(n, d, (d

a(n) = if(n%3==0, my(v=valuation(n, 2)); n>>=v; numdiv(n/3)-(!v), 0) \\ David A. Corneth, Oct 03 2018

a(n) = Sum_{d|n, dA000035(d))*A079978(d).
a(n) = A007814(A319990(n)).
a(4*n) = a(2*n). - David A. Corneth, Oct 03 2018
G.f.: Sum_{k>=1} x^(12*k-6) / (1 - x^(6*k-3)). - Ilya Gutkovskiy, Apr 14 2021
Sum_{k=1..n} a(k) = n*log(n)/6 + c*n + O(n^(1/3)*log(n)), where c = gamma(3,6) - (2 - gamma)/6 = -0.208505..., gamma(3,6) = -(psi(1/2) + log(6))/6 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A379324 Decimal expansion of log(6)^(log(5)^(log(4)^(log(3)^log(2)))).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Dec 20 2024

This is an approximation to Pi accurate to 5 digits.

			3.1415773871691905336574449813486768105453105619...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pi Approximations.
Index entries for sequences related to the number Pi

Cf. A000796, A002162, A002391, A016627, A016628, A016629.

Cf. A221185, A379276, A379321, A379322, A379323.

First[RealDigits[Log[6]^Log[5]^Log[4]^Log[3]^Log[2], 10, 100]]

Equals A016629^(A016628^(A016627^(A002391^A002162))).

A016734 Continued fraction for log(6).

Original entry on oeis.org

1, 1, 3, 1, 4, 18, 2, 330, 3, 1, 2, 1, 1, 4, 1, 4, 2, 6, 2, 2, 1, 3, 4, 5, 1, 6, 3, 1, 5, 1, 37, 1, 1, 1, 3, 1, 2, 1, 1, 2, 1, 1, 5, 4, 2, 2, 1, 37, 4, 31, 1, 1, 49, 1, 7, 1, 6, 2, 7, 2, 2, 4, 2, 6, 1, 1, 8, 1, 1, 2, 9, 1, 5, 1, 12, 1, 10, 2, 1, 87, 3, 6, 1, 4

Offset: 0

N. J. A. Sloane

			1.791759469228055000812477358... = 1 + 1/(1 + 1/(3 + 1/(1 + 1/(4 + ...)))). - _Harry J. Smith_, May 16 2009

Harry J. Smith, Table of n, a(n) for n = 0..19999
G. Xiao, Contfrac

Cf. A016629 (decimal expansion).

ContinuedFraction(Log(6)); // G. C. Greubel, Sep 15 2018

ContinuedFraction[Log[6],120] (* Harvey P. Dale, Dec 20 2014 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(log(6)); for (n=1, 20000, write("b016734.txt", n-1, " ", x[n])); } \\ Harry J. Smith, May 16 2009

Offset changed by Andrew Howroyd, Jul 10 2024

cp's OEIS Frontend

A152683 Decimal expansion of log_6 (2).

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A154009 Decimal expansion of log_6 (9).

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A279060 Number of divisors of n of the form 6*k + 1.

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A319995 Number of divisors of n of the form 6*k + 5.

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A320001 Number of proper divisors of n of the form 6*k + 1.

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A320005 Number of proper divisors of n of the form 6*k + 5.

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A173160 Decimal expansion of the constant x satisfying x^x = 6.

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A320003 Number of proper divisors of n of the form 6*k + 3.