A016635 -id:A016635

A256784

Decimal expansion of the generalized Euler constant gamma(5,12) (negated).

+10
15

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

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.

FORMULA

Equals EulerGamma/12 + 1/24*(Pi*(2-sqrt(3)) + 2*(sqrt(3)+1)*log(2) + log(3) - 4*sqrt(3) * log(sqrt(3)+1)).

Equals -(psi(5/12) + log(12))/12. - Amiram Eldar, Jan 07 2024

EXAMPLE

-0.0033729493224032970250324948185921946160340346994983953873167...

MATHEMATICA

Join[{0, 0}, RealDigits[-Log[12]/12 - PolyGamma[5/12]/12, 10, 105] // First]

PROG

(PARI) default(realprecision, 100); Euler/12 + 1/24*(Pi*(2-sqrt(3)) + 2*(sqrt(3)+1)*log(2) + log(3) - 4*sqrt(3)*log(sqrt(3)+1)) \\ G. C. Greubel, Aug 27 2018

(Magma) SetDefaultRealField(RealField(100)); R:= RealField(); EulerGamma(R)/12 + 1/24*(Pi(R)*(2-Sqrt(3)) + 2*(Sqrt(3)+1)*Log(2) + Log(3) - 4*Sqrt(3)*Log(Sqrt(3)+1)); // G. C. Greubel, Aug 27 2018

CROSSREFS

Cf. A001620 (EulerGamma), A016635, A228725 (gamma(1,2)), A256425 (gamma(1,3)), A256778-A256784 (selection of ruler-and-compass constructible gamma(r,k)).

KEYWORD

nonn,cons,easy

AUTHOR

Jean-François Alcover, Apr 10 2015

STATUS

approved

A234518

Decimal expansion of log_12 (28/13).

+10
9

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

Decimal expansion of maximal value of function alpha(n) = alpha-deviation from primality of number n = log_n(sigma(n)) - log_n(n+1) = log_n[sigma(n) / (n+1)] for n = 12, when alpha(12) = log_12(sigma(12)) - log_12(12+1) = log_12(28) - log_12(13) = log_12 (28/13) = 0,308766187…; alpha(p) = 0 for p = prime.

LINKS

Table of n, a(n) for n=0..86.

Index entries for transcendental numbers

FORMULA

Decimal expansion of (A016651-A016636) / A016635.

EXAMPLE

0,3087661875664928997884010546628878661481631771557148…

MATHEMATICA

RealDigits[Log[12, 28/13], 10, 120][[1]] (* Harvey P. Dale, Apr 17 2022 *)

PROG

(PARI) log(28/13)/log(12) \\ Michel Marcus, Dec 11 2014

CROSSREFS

Cf. A234515, A234516, A234517, A234519, A234520, A234521, A234522, A234523, A234524.

KEYWORD

nonn,cons

AUTHOR

Jaroslav Krizek, Jan 03 2014

STATUS

approved

A256783

Decimal expansion of the generalized Euler constant gamma(1,12).

+10
9

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

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.

FORMULA

Equals EulerGamma/12 + 1/24*(Pi*(2+sqrt(3)) - 2*(sqrt(3)-1)*log(2) + log(3) + 4*sqrt(3) * log(sqrt(3)+1)).

Equals Sum_{n>=0} (1/(12n+1) - 1/12*log((12n+13)/(12n+1))).

Equals -(psi(1/12) + log(12))/12. - Amiram Eldar, Jan 07 2024

EXAMPLE

0.83024988988662433938903419703214965055579639279727496201543...

MATHEMATICA

RealDigits[-Log[12]/12 - PolyGamma[1/12]/12, 10, 103] // First

PROG

(PARI) default(realprecision, 100); Euler/12 + 1/24*(Pi*(2+sqrt(3)) - 2*(sqrt(3)-1)*log(2) + log(3) + 4*sqrt(3)*log(sqrt(3)+1)) \\ G. C. Greubel, Aug 28 2018

(Magma) SetDefaultRealField(RealField(100)); R:= RealField(); EulerGamma(R)/12 + (1/24)*(Pi(R)*(2+Sqrt(3)) - 2*(Sqrt(3)-1)*Log(2) + Log(3) + 4*Sqrt(3)*Log(Sqrt(3)+1)); // G. C. Greubel, Aug 28 2018

CROSSREFS

Cf. A001620 (EulerGamma), A016635, A228725 (gamma(1,2)), A256425 (gamma(1,3)), A256778-A256784 (selection of ruler-and-compass constructible gamma(r,k)).

KEYWORD

nonn,cons,easy

AUTHOR

Jean-François Alcover, Apr 10 2015

STATUS

approved

A016740

Continued fraction for log(12).

+10
3

2, 2, 16, 15, 1, 2, 1, 1, 1, 16, 1, 12, 1, 2, 1, 6, 1, 6, 4, 3, 1, 4, 10, 3, 1, 1, 28, 1, 1, 1, 1, 2, 4, 3, 1, 2, 1, 1, 25, 3, 1, 44, 1, 3, 1, 25, 1, 17, 7, 15, 7, 15, 1, 3, 1, 2, 1, 1, 2, 7, 1, 1, 1, 4, 1, 16, 1, 4, 6, 1, 1, 1, 1, 12, 4, 7, 14, 11, 1, 1, 1, 2

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

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

G. Xiao, Contfrac

EXAMPLE

2.4849066497880003102297094... = 2 + 1/(2 + 1/(16 + 1/(15 + 1/(1 + ...)))). - Harry J. Smith, May 16 2009

MATHEMATICA

ContinuedFraction[Log[12], 100] (* G. C. Greubel, Sep 15 2018 *)

PROG

(PARI) { allocatemem(932245000); default(realprecision, 21000); x=contfrac(log(12)); for (n=1, 20000, write("b016740.txt", n-1, " ", x[n])); } \\ Harry J. Smith, May 16 2009

(Magma) ContinuedFraction(Log(12)); // G. C. Greubel, Sep 15 2018

CROSSREFS

Cf. A016635 (decimal expansion).

KEYWORD

nonn,cofr

AUTHOR

N. J. A. Sloane

EXTENSIONS

Offset changed by Andrew Howroyd, Jul 10 2024

STATUS

approved

A380005

Decimal expansion of (7/3)*log(log(12)) - exp(gamma)*log(log(12))^2, where gamma is the Euler-Mascheroni constant (A001620).

+10
1

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

Theorem 2 in Robin (1984) states that, for n >= 3, sigma(n)/n <= exp(gamma)*log(log(n)) + c/log(log(n)), with equality for n = 12, where sigma is the sum-of-divisors function (A000203) and c is the constant given by the present sequence. Cf. also Weisstein, eqs. (29) - (33).

REFERENCES

G. Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann, Journal de Mathématiques Pures et Appliquées, 63 (1984), pp. 187-213 (in French). See A073004 for a scanned copy.

LINKS

Paolo Xausa, Table of n, a(n) for n = 0..10000

Eric Weisstein's World of Mathematics, Divisor Function, see eq. (31).

FORMULA

Equals (7/3)*log(A016635) - A073004*log(A016635)^2.

EXAMPLE

0.64821364942179976272009425643532901899304479911015...

MATHEMATICA

First[RealDigits[7/3*# - Exp[EulerGamma]*#^2, 10, 100]] & [Log[Log[12]]]

CROSSREFS

Cf. A000203, A001620, A016635, A058209, A073004.

KEYWORD

nonn,cons,easy

AUTHOR

Paolo Xausa, Jan 14 2025

STATUS

approved

A323458

Decimal expansion of log(2^(1/2)*3^(1/3) / 6^(1/6)).

+10
0

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

Table of n, a(n) for n=0..99.

Index entries for transcendental numbers

FORMULA

From Jianing Song, Jan 23 2019: (Start)

Equals (1/6)*log(12) = (1/6)*A016635.

Equals (1/3)*log(2) + (1/6)*log(3) = (1/3)*A002162 + (1/6)*A002391. (End)

Equals Sum_{k>=1} H(2*k-1)/4^k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - Amiram Eldar, May 30 2021

EXAMPLE

0.4141511082980000517049515799731464734664151377572...

MATHEMATICA

RealDigits[Log[2^(1/2)*3^(1/3) / 6^(1/6)], 10, 101][[1]] (* Georg Fischer, Apr 04 2020 *)

PROG

(PARI) log( 2^(1/2)*3^(1/3) / 6^(1/6) ) \\ Charles R Greathouse IV, May 15 2019

CROSSREFS

Suggested by A230191.

Cf. A002162, A002391, A016635.

Cf. A001008, A002805.

KEYWORD

nonn,cons

AUTHOR

N. J. A. Sloane, Jan 20 2019

EXTENSIONS

a(99) corrected by Georg Fischer, Apr 04 2020

STATUS

approved