A016631 -id:A016631 - cp's OEIS frontend

A153870 Decimal expansion of log_18 (8).

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

Offset: 0

N. J. A. Sloane, Oct 30 2009

			.71943739970439433420705249473156918530453379005721169793593...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for transcendental numbers

Cf. decimal expansion of log_18(m): A152812 (m=2), A153021 (m=3), A153113 (m=4), A153444 (m=5), A153608 (m=6), A153628 (m=7), this sequence, A154017 (m=9), A154168 (m=10), A154189 (m=11), A154210 (m=12), A154400 (m=13), A154490 (m=14), A154688 (m=15), A154830 (m=16), A154898 (m=17), A155094 (m=19), A155530 (m=20), A155685 (m=21), A155784 (m=22), A155889 (m=23), A155995 (m=24).

RealDigits[Log[18, 8], 10, 100][[1]] (* Vincenzo Librandi, Sep 04 2013 *)

Equals A016631 / A016641 . - R. J. Mathar, Jun 10 2024

A154858 Decimal expansion of log_8 (17).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 30 2009

			1.3624876137501131360846886702701347846704224274482735627088...

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

Cf. decimal expansion of log_8(m): A152956 (m=3), A153204 (m=5), A153493 (m=6), A153618 (m=7), A154010 (m=9), A154159 (m=10), A154180 (m=11), A154201 (m=12), A154309 (m=13), A154468 (m=14), A154574 (m=15), this sequence, A154927 (m=18), A155060 (m=19), A155502 (m=20), A155675 (m=21), A155741 (m=22), A155827 (m=23), A155975 (m=24).

SetDefaultRealField(RealField(100)); Log(17)/Log(8); // G. C. Greubel, Aug 31 2018

RealDigits[Log[8, 17], 10, 100][[1]] (* Vincenzo Librandi, Sep 01 2013 *)

default(realprecision, 100); log(17)/log(8) \\ G. C. Greubel, Aug 31 2018

Equals A016640 / A016631 = A154847/3. - R. J. Mathar, Apr 11 2024

A188171 The number of divisors d of n of the form d == 5 (mod 8).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Mar 23 2011

a(5n) >= 1 as d=5 contributes to the count.

			a(13) = 1 because the divisor d=13 is 8+5 == 5 (mod 8).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Michael D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
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. A001826, A002325, A188169, A188170, A188172.

Cf. A001620, A016631, A354634 (psi(5/8)).

sigmamr := proc(n,m,r) local a,d ; a := 0 ; for d in numtheory[divisors](n) do if modp(d,m) = r then a := a+1 ; end if; end do: a; end proc:
A188171 := proc(n) sigmamr(n,8,5) ; end proc:

a[n_] := DivisorSum[n, 1 &, Mod[#, 8] == 5 &]; Array[a, 100] (* Amiram Eldar, Nov 25 2023 *)

A188171(n) = sumdiv(n, d, (5==(d%8)));  \\ Antti Karttunen, Jul 09 2017

A188169(n)+a(n) = A001826(n).
A188169(n)+A188170(n)-a(n)-A188172(n) = A002325(n).
G.f.: Sum_{k>=1} x^(5*k)/(1 - x^(8*k)). - Ilya Gutkovskiy, Sep 11 2019
Sum_{k=1..n} a(k) = n*log(n)/8 + c*n + O(n^(1/3)*log(n)), where c = gamma(5,8) - (1 - gamma)/8 = -0.131189..., gamma(5,8) = -(psi(5/8) + log(8))/8 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A188172 Number of divisors d of n of the form d == 7 (mod 8).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Mar 23 2011

			a(A007522(i)) = 1, any i.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Michael D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
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. A001842, A002325, A004771, A141164, A188169, A188170, A188171, A188226.

Cf. A001620, A016631, A354635 (psi(7/8)).

a188172 n = length $ filter ((== 0) . mod n) [7,15..n]
-- Reinhard Zumkeller, Mar 26 2011

sigmamr := proc(n,m,r) local a,d ; a := 0 ; for d in numtheory[divisors](n) do if modp(d,m) = r then a := a+1 ; end if; end do: a; end proc:
A188172 := proc(n) sigmamr(n,8,7) ; end proc:

Table[Count[Divisors[n],?(Mod[#,8]==7&)],{n,90}] (* _Harvey P. Dale, Mar 08 2014 *)

a(n) = sumdiv(n, d, (d % 8) == 7); \\ Amiram Eldar, Nov 25 2023

A188170(n)+a(n) = A001842(n).
A188169(n)+A188170(n)-A188171(n)-a(n) = A002325(n).
a(A188226(n))=n and a(m)<>n for m<A188226(n), n>=0; a(A141164(n))=1. - Reinhard Zumkeller, Mar 26 2011
G.f.: Sum_{k>=1} x^(7*k)/(1 - x^(8*k)). - Ilya Gutkovskiy, Sep 11 2019
Sum_{k=1..n} a(k) = n*log(n)/8 + c*n + O(n^(1/3)*log(n)), where c = gamma(7,8) - (1 - gamma)/8 = -0.212276..., gamma(7,8) = -(psi(7/8) + log(8))/8 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A256781 Decimal expansion of the generalized Euler constant gamma(1,8).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 10 2015

			0.788631390202002367443880819838976661978118204921...

G. C. Greubel, Table of n, a(n) for n = 0..10000
D. H. Lehmer, Euler constants for arithmetic progressions, Acta Arith. 27 (1975), p. 134.

Cf. A001620 (EulerGamma), A016631, A228725 (gamma(1,2)), A250129, A256425 (gamma(1,3)), A256778-A256784 (selection of ruler-and-compass constructible gamma(r,k)).

SetDefaultRealField(RealField(100)); R:= RealField(); EulerGamma(R)/8 + (1/8)*(Pi(R)/2*(Sqrt(2)+1) + Log(2) + Sqrt(2)*Log(Sqrt(2) + 1)); // G. C. Greubel, Aug 28 2018

RealDigits[-3/8*Log[2] - PolyGamma[1/8]/8, 10, 105] // First

Euler/8 + 1/8*(Pi/2*(sqrt(2)+1) + log(2) + sqrt(2)*log(sqrt(2) + 1)) \\ Michel Marcus, Apr 10 2015

Equals EulerGamma/8 + 1/8*(Pi/2*(sqrt(2)+1) + log(2) + sqrt(2)*log(sqrt(2) + 1)).
Equals Sum_{n>=0} (1/(8n+1) - 1/4*arctanh(4/(8n+5))).
Equals -(psi(1/8) + log(8))/8 = -(A250129 + A016631)/8. - Amiram Eldar, Jan 07 2024

A256782 Decimal expansion of the generalized Euler constant gamma(3,8).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 10 2015

			0.08431968843316295593904035680375480012812437382591706852303...

G. C. Greubel, Table of n, a(n) for n = 0..10000
D. H. Lehmer, Euler constants for arithmetic progressions, Acta Arith. 27 (1975), p. 134.

Cf. A001620 (EulerGamma), A016631, A228725 (gamma(1,2)), A256425 (gamma(1,3)), A256778-A256784 (selection of ruler-and-compass constructible gamma(r,k)), A354633.

SetDefaultRealField(RealField(100)); R:= RealField(); EulerGamma(R)/8 + (1/8)*(Pi(R)/2*(Sqrt(2)-1) + Log(2) - Sqrt(2)*Log(Sqrt(2)+1)); // G. C. Greubel, Aug 28 2018

Join[{0}, RealDigits[-3/8*Log[2] - PolyGamma[3/8]/8, 10, 104] // First]

default(realprecision, 100); Euler/8 + 1/8*(Pi/2*(sqrt(2)-1) + log(2) - sqrt(2)*log(sqrt(2)+1)) \\ G. C. Greubel, Aug 28 2018

Equals EulerGamma/8 + 1/8*(Pi/2*(sqrt(2)-1) + log(2) - sqrt(2)*log(sqrt(2)+1)).

Equals -(psi(3/8) + log(8))/8 = -(A354633 + A016631)/8. - Amiram Eldar, Jan 07 2024

A016687 Decimal expansion of log(64) = 6*log(2).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			4.158883083359671856503392728749059408453000806161531524724080056960361...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2.

Harry J. Smith, Table of n, a(n) for n = 1..20000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Index entries for transcendental numbers

Cf. A002162, A005900, A016492 (continued fraction), A016627, A016631.

RealDigits[Log[64],10,120][[1]] (* Harvey P. Dale, May 06 2022 *)

default(realprecision, 20080); x=log(64); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016687.txt", n, " ", d)); \\ Harry J. Smith, May 22 2009

Equals 2*A016631 = 3*A016627 = 6*A002162. - Alois P. Heinz, Aug 07 2023
From Peter Bala, Mar 05 2024: (Start)
log(64) = 4 + Sum_{n >= 1} (-1)^(n+1)/(p(n)*p(n+1)), where p(n) = n*(2*n^2 + 1)/3 = A005900.
Continued fraction: log(64) = 4 + 1/(6 + (1*2)/(6 + (2*3)/(6 + (3*4)/(6 + (4*5)/(6 + ... ))))). See A142983. Cf. A016627. (End)

A355953 Decimal expansion of (gamma + log(8)/2)/Pi.

Original entry on oeis.org

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

Offset: 0

Hugo Pfoertner, Jul 26 2022

This constant is the additive part A in the asymptotic behavior of the resistance R between two nodes in an infinite square lattice of one-ohm resistors separated by the distance vector (i,j): R(i,j) = log(sqrt(i^2+j^2))/Pi + A. From an engineering point of view, this constant summand can be regarded as a kind of near-field contribution, which contains the well-known resistance of 1/2 ohms between 2 neighboring nodes as the main part.

See, e.g., Cserti (1999) formula (33) on page 5 and Appendix B, pages 15 and 16, for a derivation of the parts of the constant.

			0.5146868528272853708539691163207527193...

József Cserti, Application of the lattice Green's function for calculating the resistance of an infinite network of resistors, American Journal of Physics, Vol. 68, No. 10 (2000), pp. 896-906; arXiv preprint, arXiv:cond-mat/9909120 [cond-mat.mes-hall], 1999-2000.

Cf. A001620, A016631, A355955, A355954 (similar for triangular lattice).

Cf. A355565, A355566, A355567 (exact solutions for small distances).

RealDigits[(EulerGamma + Log[8]/2)/Pi, 10, 120][[1]] (* Amiram Eldar, Jun 18 2023 *)

PARI
```
(Euler + log(8)/2)/Pi
```

A262023 Decimal expansion of 3*log(2)/2.

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Sep 08 2015

This is the limit of the reordered alternating harmonic series 1 + 1/3 - 1/2 + 1/5 + 1/7 - 1/4 + 1/9 + 1/11 - 1/6 + ... + ... - ..., with partial sums given in A262031/A262022. This shows that the alternating harmonic series is conditionally convergent. For original references on such series see A262031.

			1.039720770839917964125848182187264852113250201540382881181020014240...

Srinivasa Ramanujan, Question 260, Journal of the Indian Mathematical Society, Vol. 3 (1911), p. 43.
Eric Weisstein's World of Mathematics, Conditional Convergence.
Index entries for transcendental numbers.

Cf. A002162, A016631, A016655, A262031, A262022.

3*Log(2)/2; // Vincenzo Librandi, Jan 01 2025

First@ RealDigits@ N[3 Log[2]/2, 120] (* Michael De Vlieger, Jul 26 2016 *)

3*log(2)/2 \\ Michel Marcus, Sep 13 2015

Equals 3*A002162/2.
Equals A016631/2.
3*log(2)/2 = (3/2)*Sum_{n>=1} (-1)^(n+1)/n = Sum_{n>=1} ((-1)^(n+1)/n + (-1)^(n+1)/(2*n)) = A002162 + (A016655/10). - Terry D. Grant, Jul 24 2016
Equals 1 + Sum_{k>=1} 2/((4*k)^3 - 4*k) (Ramanujan, 1911). - Amiram Eldar, Jan 01 2025

A016736 Continued fraction for log(8).

Original entry on oeis.org

2, 12, 1, 1, 2, 2, 1, 8, 1, 4, 1, 31, 1, 8, 1, 4, 1, 6, 1, 3, 4, 2, 1, 1, 1, 2, 1, 4, 1, 1, 2, 6, 1, 1, 1, 7, 1, 1, 2, 4, 1, 1, 1, 3, 3, 1, 2, 55, 1, 3, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 3, 3, 41, 1, 3, 1, 2, 2, 12, 1, 5, 2, 5, 2, 2, 1, 3, 1, 10, 4, 1, 12, 2, 2, 1

Offset: 0

N. J. A. Sloane

			2.07944154167983592825169636... = 2 + 1/(12 + 1/(1 + 1/(1 + 1/(2 + ...)))). - _Harry J. Smith_, May 16 2009

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

Cf. A016631 (decimal expansion).

ContinuedFraction(3*Log(2)); // G. C. Greubel, Sep 15 2018

ContinuedFraction[3*Log[2], 100] (* G. C. Greubel, Sep 15 2018 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(log(8)); for (n=1, 20000, write("b016736.txt", n-1, " ", x[n])); } \\ Harry J. Smith, May 16 2009

Offset changed by Andrew Howroyd, Jul 10 2024

cp's OEIS Frontend

A153870 Decimal expansion of log_18 (8).

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A154858 Decimal expansion of log_8 (17).

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Magma

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A188171 The number of divisors d of n of the form d == 5 (mod 8).

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A188172 Number of divisors d of n of the form d == 7 (mod 8).

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A256781 Decimal expansion of the generalized Euler constant gamma(1,8).

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Magma

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A256782 Decimal expansion of the generalized Euler constant gamma(3,8).

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Magma

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PARI

Formula

A016687 Decimal expansion of log(64) = 6*log(2).

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