A001620 -id:A001620 - cp's OEIS frontend

A059856 Write down decimal expansion of Euler-Mascheroni constant gamma (A001620); divide up into minimal chunks so that chunks have increasing length and do not begin with zero.

5, 77, 215, 66490, 1532860, 6065120900, 82402431042, 159335939923, 5988057672348, 84867726777664, 670936947063291, 7467495146314472, 498070824809605040, 1448654283622417399, 76449235362535003337

Offset: 0

Jason Earls, Feb 27 2001

			0.5772156649015328606065120900824024310421593359399235...

Cf. A001620, A002852, A033091, A098967.

More terms from Tracy Poff (tracy.poff(AT)gmail.com), Apr 15 2005

A089044 Numbers n such that abs(d(n) - log(n) + 1 - 2*gamma) is a decreasing sequence, where d(n) is the number of divisors A000005(n) and gamma is Euler's constant A001620.

Original entry on oeis.org

1, 3, 5, 7, 46, 2514, 2522, 2526, 2534, 2536, 2542, 2546, 2553, 2555, 18873, 139454, 139475, 7614005, 7614010, 7614015, 7614022, 7614030, 7614033, 7614034, 7614056, 7614062, 7614066, 7614069, 7614079, 7614082, 7614086, 7614087, 7614088

Offset: 1

Leroy Quet and Hugo Pfoertner, Dec 02 2003

nonn

			a(5)=46 because d(46) - log(46) + 1 - 2*0.5772156649... = 0.016927274... is less than
abs(d(7) - log(7) + 1 - 2*0.5772156649...) = abs(-0.100341479...) with d(46)=4 and d(7)=2.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Theorem 320.

Hugo Pfoertner, Table of n, a(n) for n = 1..7613
Leroy Quet, Two number-theoretical limits (& bonus sum). Thread in NG sci.math, Oct 30 2003.
Eric Weisstein's World of Mathematics, Euler-Mascheroni Constant

Cf. A000005 = number of divisors of n, A001620 = Euler's constant gamma, A089084.

f[n_] := N[ Abs[ DivisorSigma[0, n] - Log@ n + 1 - 2 EulerGamma], 32]; k = 1; lst = {}; mx = Infinity; While[k < 8000000, a = f@k; If[a < mx, mx = a; AppendTo[lst, k]]; k++]; lst (* Robert G. Wilson v, Dec 11 2017 *)

d=1.0;n=0;\
for(j=2,16,kmin=round(exp(j-2*Euler+1-2*d));kmax=round(exp(j-2*Euler+1+2*d));\
for(k=kmin,kmax,dd=abs(numdiv(k)-log(k)+1-2*Euler);\
if(ddHugo Pfoertner, Dec 08 2017

Terms beyond a(5) from Hans Havermann, Dec 02 2003

A089084 Numbers n such that abs ( (sum_m (m=1..n) d(m)) / n - log(n) - 2*gamma + 1) is a decreasing sequence, where d(m) is the number of divisors A000005(m) and gamma is Euler's constant A001620.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 17, 19, 23, 47, 89, 125, 131, 203, 219, 455, 1475, 2867, 4649, 7291, 36893, 378878, 517914, 693028, 923373, 1835331, 3147909, 3356513, 3506524, 6782094, 20454813, 25494256, 27802807, 28081980, 47214722, 176344865

Offset: 1

Hugo Pfoertner, Dec 04 2003

nonn

For references and links see A006218 and A089044.

Hugo Pfoertner, Table of n, a(n) for n = 1..46

Cf. A000005, A001620, A006218, A089044.

s=0;r=2;for(k=1,10^7,s=s+numdiv(k);t=abs(s/k-log(k)-2*Euler+1);if(abs(t)Hugo Pfoertner, Aug 30 2018

Terms a(12) and beyond from Hans Havermann

A104015 Binary expansion of Euler's constant (or Euler-Mascheroni constant) gamma (A001620).

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0

Offset: 0

Bryan Jacobs (bryanjj(AT)gmail.com), Mar 31 2005

			.1001001111000100011001111110...

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A001620.

RealDigits[EulerGamma,2,120][[1]] (* Harvey P. Dale, May 25 2018 *)

binary(Euler)[2] \\ Michel Marcus, Dec 14 2017

A104938 Primes from merging of 4 successive digits in decimal expansion of the Euler-Mascheroni constant A001620.

Original entry on oeis.org

3359, 3593, 5939, 9923, 8677, 2677, 6709, 6947, 6329, 2917, 4951, 1447, 4283, 2417, 6449, 5003, 3733, 3767, 7673, 9491, 2039, 5323, 6211, 4793, 7937, 7057, 3547, 6043, 6733, 7331, 3313, 1399, 7541, 5413, 4139, 8423, 4877, 8431, 3109, 1093, 9973, 3613

Offset: 1

Andrew G. West (WestA(AT)wlu.edu), Mar 29 2005

Leading zeros are not permitted, so each term is 4 digits in length.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Jon Borwein, 170,000 digits of Gamma [Wayback Machine copy]
Eric Weisstein's World of Mathematics, Euler-Mascheroni Constant.

Cf. A198778.

Cf. A103773, A103789, A103793, A103808 - A103812, A104824- A104826, A104843 - A104850, A198161 - A198177, A198776 - A198784. - Harvey P. Dale, Oct 29 2011

Digits := 420 ;
for sh from 3 do
        p := floor(gamma*10^sh) mod 10000 ;
        if isprime(p) and p > 999 then
                printf("%d,",p);
        end if;
end do: # R. J. Mathar, Oct 31 2011

egp[len_]:=Module[{egterms=FromDigits/@Partition[RealDigits[EulerGamma, 10, 1000][[1]],len,1]},Select[egterms,IntegerLength[#]==len&&PrimeQ[#]&]]; egp[4] (* Harvey P. Dale, Oct 29 2011 *)

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

Offset changed from 0 to 1 by Vincenzo Librandi, Apr 21 2013

A131447 Numerators of n-th approximation of factorial (also called harmonic) expansion of Euler's constant gamma (A001620).

Original entry on oeis.org

0, 1, 1, 13, 23, 83, 2909, 23273, 3491, 3491, 11520301, 30720803, 30720803, 50320675319, 68619102709, 3019240519199, 4666098984217, 1847775197749939, 23405152504832563, 1404309150289953793, 9830164052029676557

Offset: 1

Wolfdieter Lang, Aug 07 2007

Rationals are in lowest terms.

			Rationals r(n): [0, 1/2, 1/2, 13/24, 23/40, 83/144, 2909/5040, 23273/40320, 3491/6048, ...]

J. Havil, Gamma, (in German), Springer, 2007, p. 118; Gamma: Exploring Euler's Constant, Princeton Univ. Press, 2003.

Wolfdieter Lang, Rationals and values.

Cf. A001620, A096622, A131448 (denominators)

a(n) = numerator(r(n)), with r(n) = Sum_{k=1..n} b(k)/k!, and b(k) = A096622(k) (factorial expansion of gamma).

A175794 a(n) = Sum_{k=1..n} (-1)^A001620(k).

Original entry on oeis.org

-1, -2, -3, -2, -3, -4, -3, -2, -1, -2, -1, -2, -3, -4, -3, -2, -1, 0, 1, 2, 3, 2, 1, 2, 3, 2, 3, 4, 5, 6, 7, 8, 9, 10, 9, 8, 9, 10, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 2, 1, 0, -1, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 2, 3, 4, 5, 6, 5, 4, 5, 6, 5, 4, 3, 4, 5, 6, 7, 6, 7, 6, 5, 6, 5, 6, 5, 6, 7, 6, 7, 6, 5, 4, 5

Offset: 1

Michel Lagneau, Sep 06 2010

			a(6) = -4 is in the sequence because -4 = (-1)^5 + (-1)^7 + (-1)^7 + (-1)^2 + (-1)^1 + (-1)^5

G. C. Greubel, Table of n, a(n) for n = 1..10000

A001620 is the decimal expansion of Euler's constant (or Euler-Mascheroni constant) gamma.

with(numtheory):T:=array(1..201): Digits:=200:nn:=10^200:a:=floor(evalf(gamma(0))*nn): n:=a:l:=length(n):n0:=n:s:=0:for m from 1 to l do:q:=n0:u:=irem(q,10):v:=iquo(q,10):n0:=v :T[m]:=u:od: for p from l to 1 by -1 do:s:=s+(-1)^T[p]: printf(`%d, `, s):od:

Rest@ FoldList[ Plus, 0, (-1)^First@ RealDigits[EulerGamma, 10, 200]] (* or *) Accumulate[(-1)^RealDigits[EulerGamma,10,100][[1]]] (* Harvey P. Dale, May 11 2014 *)

A195606 Numerator of floor(gamma*10^n)/10^n, where gamma=A001620 is the Euler-Mascheroni constant.

Original entry on oeis.org

0, 1, 57, 577, 1443, 57721, 115443, 1443039, 28860783, 36075979, 5772156649, 5772156649, 577215664901, 1154431329803, 57721566490153, 144303916225383, 180379895281729, 28860783245076643, 28860783245076643, 2886078324507664303

Offset: 0

M. F. Hasler, following a suggestion by Eric Angelini, Sep 21 2011

nonn

Numerator of the decimal fraction of gamma=0.5772... truncated to a given number of decimal places.

			a(3) = 577 is the numerator of 0.577 = 577/1000.
a(4) = 1443 is the numerator of 0.5772 = 5772/10000 = 1443/2500.

G. C. Greubel, Table of n, a(n) for n = 0..995

R:=RealField(100); [Numerator(Floor(EulerGamma(R)*10^n)/10^n): n in [0..50]]; // G. C. Greubel, Aug 27 2018

Numerator[Table[Floor[EulerGamma*10^n]/10^n, {n, 0, 50}]] (* G. C. Greubel, Aug 27 2018 *)

a(n,c=Euler)=numerator(c\.1^n/10^n)  \\ M. F. Hasler, Sep 21 2011

A213440 Decimal expansion of 1 + log(gamma), where gamma is Euler's constant A001620.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Jun 11 2012

			0.4504606870183551776623382311970922116693010187369352...

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, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A001620, A002389.

R:= RealField(100); 1 + Log(EulerGamma(R)); // G. C. Greubel, Aug 27 2018

RealDigits[1 + Log[EulerGamma], 10, 100][[1]] (* G. C. Greubel, Aug 27 2018 *)

default(realprecision, 100); 1 + log(Euler) \\ G. C. Greubel, Aug 27 2018

A215722 Decimal expansion of Pi*(3 - gamma)/32, where gamma is Euler's constant A001620.

Original entry on oeis.org

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

Offset: 0

Charles R Greathouse IV, Aug 22 2012

Volchkov shows that this is equal to integral(t=0..oo, (1-12*t^2)/(1+4*t^2)^3) * integral(s=1/2..oo, log |zeta(s + i*t)|) if and only if the Riemann hypothesis holds.

			0.237856295886805506742962363080233394796370125523522395446521428085...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.2, p. 42.

G. C. Greubel, Table of n, a(n) for n = 0..10000
V. V. Volchkov, On an equality equivalent to the Riemann hypothesis, Ukrainian Mathematical Journal 47:3 (1995), pp. 491-493.

R:= RealField(100); Pi(R)*(3 - EulerGamma(R))/32; // G. C. Greubel, Aug 27 2018

RealDigits[Pi*(3 - EulerGamma)/32, 10, 100][[1]] (* G. C. Greubel, Aug 27 2018 *)

PARI
```
Pi*(3-Euler)/32
```

cp's OEIS Frontend

A059856 Write down decimal expansion of Euler-Mascheroni constant gamma (A001620); divide up into minimal chunks so that chunks have increasing length and do not begin with zero.

Views

Author

Keywords

Examples

Crossrefs

Extensions

A089044 Numbers n such that abs(d(n) - log(n) + 1 - 2*gamma) is a decreasing sequence, where d(n) is the number of divisors A000005(n) and gamma is Euler's constant A001620.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A089084 Numbers n such that abs ( (sum_m (m=1..n) d(m)) / n - log(n) - 2*gamma + 1) is a decreasing sequence, where d(m) is the number of divisors A000005(m) and gamma is Euler's constant A001620.

Views

Author

Keywords

References

Links

Crossrefs

Programs

PARI

Extensions

A104015 Binary expansion of Euler's constant (or Euler-Mascheroni constant) gamma (A001620).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A104938 Primes from merging of 4 successive digits in decimal expansion of the Euler-Mascheroni constant A001620.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A131447 Numerators of n-th approximation of factorial (also called harmonic) expansion of Euler's constant gamma (A001620).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

A175794 a(n) = Sum_{k=1..n} (-1)^A001620(k).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A195606 Numerator of floor(gamma*10^n)/10^n, where gamma=A001620 is the Euler-Mascheroni constant.

Views

Author

Keywords

Comments

Examples

Links

Programs