A001620 -id:A001620 - cp's OEIS frontend

A097663 Decimal expansion of the constant 3*exp(psi(1/3) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

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

Offset: 0

Paul D. Hanna, Aug 25 2004

This constant appears in Benoit Cloitre's generalized Euler-Gauss formula for the Gamma function (see Cloitre link) and is involved in the exact determination of asymptotic limits of certain order-3 linear recursions with varying coefficients (see A097677 for example).

			0.23311909318456411730537562326544289574460858702592456414096...

A. M. Odlyzko, Linear recurrences with varying coefficients, in Handbook of Combinatorics, Vol. 2, R. L. Graham, M. Grotschel and L. Lovasz, eds., Elsevier, Amsterdam, 1995, pp. 1135-1138.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Benoit Cloitre, On a generalization of Euler-Gauss formula for the Gamma function, preprint 2004.
Xavier Gourdon and Pascal Sebah, Introduction to the Gamma Function.
Andrew Odlyzko, Asymptotic enumeration methods, in Handbook of Combinatorics, vol. 2, 1995, pp. 1063-1229.

Cf. A001620, A047787, A097664, A097665-A097676.

SetDefaultRealField(RealField(100)); R:= RealField(); Exp(-Pi(R)/Sqrt(12))/Sqrt(3); // G. C. Greubel, Sep 07 2018

RealDigits[1/Sqrt[3]*E^(-Pi/Sqrt[12]), 10, 105][[1]] (* Robert G. Wilson v, Aug 28 2004 *)

PARI
```
3*exp(psi(1/3)+Euler)
```

Equals exp(-Pi/sqrt(12))/sqrt(3).

More terms from Robert G. Wilson v, Aug 28 2004

Offset corrected by R. J. Mathar, Feb 05 2009

A097676 Decimal expansion of the constant 8*exp(psi(7/8) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

Original entry on oeis.org

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

Offset: 1

Paul D. Hanna, Aug 25 2004

This constant appears in Benoit Cloitre's generalized Euler-Gauss formula for the Gamma function (see Cloitre link) and is involved in the exact determination of asymptotic limits of certain order-8 linear recursions with varying coefficients (see A097682 for example).

			c = 6.37663248941667785500176259382510790626743532678646216767306...

A. M. Odlyzko, Linear recurrences with varying coefficients, in Handbook of Combinatorics, Vol. 2, R. L. Graham, M. Grotschel and L. Lovasz, eds., Elsevier, Amsterdam, 1995, pp. 1135-1138.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Benoit Cloitre, On a generalization of Euler-Gauss formula for the Gamma function, preprint 2004.
Xavier Gourdon and Pascal Sebah, Introduction to the Gamma Function.
Andrew Odlyzko, Asymptotic enumeration methods, in Handbook of Combinatorics, vol. 2, 1995, pp. 1063-1229.

Cf. A097663-A097675.

SetDefaultRealField(RealField(100)); R:= RealField();
(1+Sqrt(2))^(-Sqrt(2))/2*Exp(Pi(R)/2*(1+Sqrt(2))); // G. C. Greubel, Sep 07 2018

RealDigits[(1 + Sqrt[2])^(-Sqrt[2])/2E^(Pi/2*(1 + Sqrt[2])), 10, 105][[1]] (* Robert G. Wilson v, Aug 27 2004 *)

PARI
```
8*exp(psi(7/8)+Euler)
```

c = (1+sqrt(2))^(-sqrt(2))/2*exp(Pi/2*(1+sqrt(2))).

More terms from Robert G. Wilson v, Aug 27 2004

A195189 Denominators of a sequence leading to gamma = A001620.

Original entry on oeis.org

2, 24, 72, 2880, 800, 362880, 169344, 29030400, 9331200, 4790016000, 8673280, 31384184832000, 6181733376000, 439378587648000, 10346434560000, 512189896458240000, 265423814656, 14148260909088768000, 2076423318208512000, 96342919523794944000000, 74538995631567667200000

Offset: 0

Paul Curtz, Sep 11 2011

gamma = 1/2 + 1/24 + 1/72 + 19/2880 + 3/800 + 863/362880 + 275/169344 + ... = (A002206 unsigned=reduced A141417(n+1)/A091137(n+1))/a(n) is an old formula based on Gregory's A002206/A002207.

This formula for Euler's constant was discovered circa 1780-1790 by the Italian mathematicians Gregorio Fontana (1735-1803) and Lorenzo Mascheroni (1750-1800), and was subsequently rediscovered several times (in particular, by Ernst Schröder in 1879, Niels E. Nørlund in 1923, Jan C. Kluyver in 1924, Charles Jordan in 1929, Kenter in 1999, and Victor Kowalenko in 2008). For more details, see references below. - Iaroslav V. Blagouchine, May 03 2015

			a(0)=1*2, a(1)=2*12, a(2)=3*24, a(3)=4*720.

G. C. Greubel, Table of n, a(n) for n = 0..440
Iaroslav V. Blagouchine, A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations, Journal of Number Theory (Elsevier), vol. 148, pp. 537-592 and vol. 151, pp. 276-277, 2015. arXiv version.
Iaroslav V. Blagouchine, Expansions of generalized Euler's constants into the series of polynomials in 1/pi^2 and into the formal enveloping series with rational coefficients only, Journal of Number Theory (Elsevier), vol. 158, pp. 365-396, 2016. arXiv version.
M. Coffey and J. Sondow, Rebuttal of Kowalenko's paper as concerns the irrationality of Euler's constant, Acta Appl. Math., 121 (2012), 1-3.
J. C. Kluyver, Euler's constant and natural numbers, Proc. Kon. Ned. Akad. Wet., 27(1-2) (1924), 142-144.

Cf. A001620, A002206, A002207, A091137, A141417.

g[n_]:=Sum[StirlingS1[n,l]/(l+1),{l,1,n}]/(n*n!); a[n_]:=Denominator[g[n]]; Table[a[n],{n,1,30}] (* Iaroslav V. Blagouchine, May 03 2015 *)
g[n_] := Sum[ BernoulliB[j]/j * StirlingS1[n, j-1], {j, 1, n+1}] / n! ; a[n_] := (n+1)*Denominator[g[n]]; Table[a[n], {n, 0, 20}]
(* or *) max = 20; Denominator[ CoefficientList[ Series[ 1/Log[1 + x] - 1/x, {x, 0, max}], x]]*Range[max+1] (* Jean-François Alcover, Sep 04 2013 *)

a(n) = (n+1) * A002207(n).

More terms from Jean-François Alcover, Sep 04 2013

A058209 a(n) = floor( exp(gamma) n log log n ) - sigma(n), where gamma is Euler's constant (A001620) and sigma(n) is sum of divisors of n (A000203).

Original entry on oeis.org

-5, -4, -5, -2, -6, 0, -5, -1, -4, 5, -9, 7, 0, 2, -2, 13, -5, 16, -3, 9, 8, 22, -11, 21, 12, 17, 4, 32, -7, 36, 7, 25, 22, 31, -10, 46, 27, 34, 2, 53, 2, 57, 20, 29, 37, 64, -9, 61, 28, 52, 29, 76, 13, 63, 18, 61, 54, 87, -18, 91, 60, 55, 35, 81, 24, 103, 48, 81, 36, 111, -9, 115

Offset: 2

N. J. A. Sloane, Nov 30 2000

Theorem (G. Robin): exp(gamma) n log log n - sigma(n) is positive for all n >= 5041 if and only if the Riemann Hypothesis is true.

Note that a(n) <= exp(gamma) n log log n - sigma(n) < a(n) + 1.

D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section III.2.2.b.
G. Robin, Grandes valeurs de la fonction somme des diviseurs et hypothese de Riemann, J. Math. Pures Appl. 63 (1984), 187-213.

T. D. Noe, Table of n, a(n) for n = 2..10000
G. Caveney, J.-L. Nicolas, and J. Sondow, Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis, Integers 11 (2011), #A33.
G. Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, Ramanujan J., 29 (2012), 359-384.

Cf. A000203, A001620, A057641, A057642, A058210.

with(numtheory); Digits := 100; g := evalf(gamma); [seq( floor(exp(g)*n*log(log(n)))-sigma[1](n), n=2..80)];

a[n_] := Floor[Exp[EulerGamma] n*Log[Log[n]]] - DivisorSigma[1, n]; Array[a,100,2] (* Jean-François Alcover, May 04 2011 *)

a(n)=floor( exp(Euler)*n*log(log(n)) - sigma(n)) \\ Charles R Greathouse IV, Feb 08 2017

Statement of Robin's theorem corrected by Jonathan Sondow, May 30 2011

A198778 Primes from merging of 4 successive digits in decimal expansion of Euler-Mascheroni constant A001620.

Original entry on oeis.org

577, 421, 3359, 3593, 5939, 9923, 8677, 2677, 6709, 6947, 6329, 2917, 4951, 1447, 401, 4283, 2417, 6449, 5003, 3733, 3767, 7673, 9491, 2039, 853, 5323, 6211, 4793, 7937, 857, 7057, 29, 3547, 6043, 587, 6733, 7331, 3313, 1399, 7541, 5413, 4139, 8423, 4877, 503, 8431, 3109, 1093, 9973, 3613, 8893, 8933, 17, 7247

Offset: 1

Harvey P. Dale, Oct 29 2011

In contrast to A104938, leading zeros are allowed here, which explains the terms having fewer than 4 digits; e.g., a(32)=29 comes from consecutive digits "...0029..." starting at the 268th decimal digit of gamma (if the initial "0." counts as the first digit). - M. F. Hasler, Oct 31 2011

			The first four decimal digits of gamma = 0.5772... form the prime 577=a(1).

Cf. A103773, A103789, A103793, A103808-A103812, A104824-A104826, A104843-A104850, A198161-A198177, A198776-A198784.

Digits := 420 ;
for sh from 3 do
        p := floor(gamma*10^sh) mod 10000 ;
        if isprime(p) then
                printf("%d,",p);
        end if;
end do: # R. J. Mathar, Oct 31 2011

(* see A104938 for Mmca code *)
Join[{577},Select[FromDigits/@Partition[RealDigits[EulerGamma,10,1000][[1]],4,1],PrimeQ]] (* Harvey P. Dale, May 07 2019 *)

L=10^4;for(i=3,999,isprime(p=Euler\.1^i%L)&print1(p",")) \\ M. F. Hasler, Oct 31 2011

A002389 Decimal expansion of -log(gamma), where gamma is Euler's constant A001620.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

From Peter Bala, Aug 24 2025: (Start)
By definition, the Euler-Mascheroni constant gamma = lim_{n -> oo} s(n), where s(n) = Sum_{k = 1..n} 1/k - log(n). The convergence is slow. For example, s(50) = 0.5(87...) is only correct to 1 decimal digit. Let S(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(n, k)*binomial(n+k, k)*s(n+k). Elsner shows that S(n) converges to gamma much more rapidly. For example, S(50) = 0.57721566490153286060651209008(02...) gives gamma correct to 29 decimal digits.
Define E(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(n, k)*binomial(n+k, k)*log(s(n+k)). Then it appears that E(n) converges rapidly to log(gamma). For example, E(50) = -0.549539312981644822337661768802(88...) gives log(gamma) correct to 30 decimal digits. Cf. A073004. (End)

			.549539312981644822337661768802907788330698981263...

W. E. Mansell, Tables of Natural and Common Logarithms. Royal Society Mathematical Tables, Vol. 8, Cambridge Univ. Press, 1964, p. XVIII.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Ivan Panchenko, Table of n, a(n) for n = 0..1000
C. Elsner, On a sequence transformation with integral coefficients for Euler's constant, Proc. Amer. Math. Soc., Vol. 123 (1995), Number 5, pp. 1537-1541.
Simon Plouffe, -log(gamma) to 10000 digits
Simon Plouffe, -log(gamma) to 1024 digits

Cf. A001620, A073004, A155969, A213440.

SetDefaultRealField(RealField(100)); R:= RealField(); -Log(EulerGamma(R)); // G. C. Greubel, Sep 07 2018

RealDigits[-Log[EulerGamma], 10, 100][[1]] (* G. C. Greubel, Sep 07 2018 *)

-log(Euler) \\ Michel Marcus, Mar 11 2013

A097665 Decimal expansion of the constant 4*exp(psi(1/4) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

Original entry on oeis.org

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

Offset: 0

Paul D. Hanna, Aug 25 2004

This constant appears in Benoit Cloitre's generalized Euler-Gauss formula for the Gamma function (see Cloitre link) and is involved in the exact determination of asymptotic limits of certain order-4 linear recursions with varying coefficients (see A097679 for example).

			c = 0.10393978817538095427347780991748938501693892081588480403756...

A. M. Odlyzko, Linear recurrences with varying coefficients, in Handbook of Combinatorics, Vol. 2, R. L. Graham, M. Grotschel and L. Lovasz, eds., Elsevier, Amsterdam, 1995, pp. 1135-1138.

G. C. Greubel, Table of n, a(n) for n = 0..2500
Benoit Cloitre, On a generalization of Euler-Gauss formula for the Gamma function, preprint 2004.
Andrew Odlyzko, Asymptotic enumeration methods, in Handbook of Combinatorics, vol. 2, 1995, pp. 1063-1229.
Xavier Gourdon and Pascal Sebah, Introduction to the Gamma Function.

Cf. A097663-A097664, A097666-A097676.

RealDigits[1/2*E^(-Pi/2), 10, 105][[1]] (* Robert G. Wilson v, Aug 28 2004 *)

PARI
```
4*exp(psi(1/4)+Euler)
```

c = 1/2*exp(-Pi/2).

More terms from Robert G. Wilson v, Aug 28 2004

A038128 Beatty sequence for Euler's constant (A001620).

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 4, 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, 21, 22, 23, 23, 24, 24, 25, 25, 26, 27, 27, 28, 28, 29, 30, 30, 31, 31, 32, 32, 33, 34, 34, 35, 35, 36, 36, 37, 38, 38, 39, 39, 40, 40, 41, 42, 42, 43

Offset: 0

Felice Russo

Karl V. Keller, Jr., Table of n, a(n) for n = 0..100000
Index entries for sequences related to Beatty sequences

[Floor((n*EulerGamma(RealField(100)))) : n in [0..100]]; // Vincenzo Librandi, Aug 15 2015

Mathematica
```
Table[Floor[n*EulerGamma], {n, 0, 100}]
```

a(n) = floor(n*Euler); \\ Michel Marcus, Oct 24 2015

a(n) = floor(n*0.5772156649...). - Typo fixed by Karl V. Keller, Jr., Jul 27 2015

More terms from Leah Frazee (s1166278(AT)Cedarville.edu)

a(0) term added by T. D. Noe, Mar 27 2011

A058210 a(n) = floor( exp(gamma) n log log n ), where gamma is Euler's constant (A001620).

Original entry on oeis.org

-2, 0, 2, 4, 6, 8, 10, 12, 14, 17, 19, 21, 24, 26, 29, 31, 34, 36, 39, 41, 44, 46, 49, 52, 54, 57, 60, 62, 65, 68, 70, 73, 76, 79, 81, 84, 87, 90, 92, 95, 98, 101, 104, 107, 109, 112, 115, 118, 121, 124, 127, 130, 133, 135, 138, 141, 144, 147, 150

Offset: 2

N. J. A. Sloane, Nov 30 2000

sign

Theorem (G. Robin): exp(gamma) n log log n > sigma(n) for all n >= 5041 if and only if the Riemann Hypothesis is true.

Note that a(n) <= exp(gamma) n log log n < a(n) + 1.

D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section III.2.2.b.
G. Robin, Grandes valeurs de la fonction somme des diviseurs et hypothese de Riemann, J. Math. Pures Appl. 63 (1984), 187-213.

G. C. Greubel, Table of n, a(n) for n = 2..1000
G. Caveney, J.-L. Nicolas, and J. Sondow, Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis, Integers 11 (2011), #A33.
G. Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, Ramanujan J., 29 (2012), 359-384.

See A058209.

Cf. A001620.

a:= n-> floor(exp(gamma)*n*log(log(n))):
seq(a(n), n=2..60);  # Alois P. Heinz, Oct 18 2022

Table[Floor[Exp[EulerGamma]*n*Log[Log[n]]], {n,2,50}] (* G. C. Greubel, Dec 31 2016 *)

Statement of Robin's theorem corrected by Jonathan Sondow, May 30 2011

A059555 Beatty sequence for 1 + gamma A001620.

Original entry on oeis.org

1, 3, 4, 6, 7, 9, 11, 12, 14, 15, 17, 18, 20, 22, 23, 25, 26, 28, 29, 31, 33, 34, 36, 37, 39, 41, 42, 44, 45, 47, 48, 50, 52, 53, 55, 56, 58, 59, 61, 63, 64, 66, 67, 69, 70, 72, 74, 75, 77, 78, 80, 82, 83, 85, 86, 88, 89, 91, 93, 94, 96, 97, 99, 100, 102, 104, 105, 107, 108

Offset: 1

Mitch Harris, Jan 22 2001

Let r = gamma (the Euler constant, 0.5772...). When {k*r, k >= 1} is jointly ranked with the positive integers, A059555(n) is the position of n and A059556(n) is the position of n*r. - Clark Kimberling, Oct 21 2014

Harry J. Smith, Table of n, a(n) for n = 1..2000
Fraenkel, Aviezri S.; Levitt, Jonathan; Shimshoni, Michael; Characterization of the set of values f(n)=[n alpha], n=1,2,..., Discrete Math. 2 (1972), no. 4, 335-345.
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences

Beatty complement is A059556.

R:=RealField(100); [Floor((1+EulerGamma(R))*n): n in [1..100]]; // G. C. Greubel, Aug 27 2018

A001620 := proc(n)
        floor((1+gamma)*n) ;
end proc:
seq(A001620(n),n=1..50) ; # R. J. Mathar, Nov 11 2011

t = N[Table[k*EulerGamma, {k, 1, 200}]]; u = Union[Range[200], t]
Flatten[Table[Flatten[Position[u, n]], {n, 1, 100}]]  (* A059556 *)
Flatten[Table[Flatten[Position[u, t[[n]]]], {n, 1, 100}]] (* A059555 *)
(* Clark Kimberling, Oct 21 2014 *)

{ default(realprecision, 100); b=1 + Euler; for (n = 1, 2000, write("b059555.txt", n, " ", floor(n*b)); ) } \\ Harry J. Smith, Jun 28 2009

a(n) = n + A038128(n).

cp's OEIS Frontend

A097663 Decimal expansion of the constant 3*exp(psi(1/3) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A097676 Decimal expansion of the constant 8*exp(psi(7/8) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A195189 Denominators of a sequence leading to gamma = A001620.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A058209 a(n) = floor( exp(gamma) n log log n ) - sigma(n), where gamma is Euler's constant (A001620) and sigma(n) is sum of divisors of n (A000203).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A198778 Primes from merging of 4 successive digits in decimal expansion of Euler-Mascheroni constant A001620.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

A002389 Decimal expansion of -log(gamma), where gamma is Euler's constant A001620.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI