A242624 -id:A242624 - cp's OEIS frontend

A085361 Decimal expansion of the number c = Sum_{n>=1} (zeta(n+1)-1)/n.

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

Offset: 0

Eric W. Weisstein, Jun 25 2003

The Alladi-Grinstead constant (A085291) is exp(c-1).

			0.78853056591150896106027632345455466647274966822328164975515640230178...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.8.1 Alternative representations [of real numbers], p. 62.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 528 and 538.
Sofia Kalpazidou, Khintchine's constant for Lüroth representation, Journal of Number Theory, Vol. 29, No. 2 (June 1988), pp. 196-205.
Eric Weisstein's World of Mathematics, Alladi-Grinstead Constant.
Eric Weisstein's World of Mathematics, Convergence Improvement.

Cf. A002210, A085291, A242624, A244109, A245254.

SetDefaultRealField(RealField(120)); L:=RiemannZeta(); (&+[(Evaluate(L,n+1)-1)/n: n in [1..1000]]); // G. C. Greubel, Nov 15 2018

evalf(sum((Zeta(n+1)-1)/n, n=1..infinity), 120); # Vaclav Kotesovec, Dec 11 2015
evalf(Sum(-(-1)^k*Zeta(1, k), k = 2..infinity), 120); # Vaclav Kotesovec, Jun 18 2021

Sum[(-1+Zeta[1+n])/n,{n,Infinity}]
NSum[Log[k]/(k*(k+1)), {k, 1, Infinity}, WorkingPrecision -> 120, NSumTerms ->5000, Method -> {NIntegrate, MaxRecursion -> 100}] (* Vaclav Kotesovec, Dec 11 2015 *)

suminf(n=1,(zeta(n+1)-1-2^(-n-1))/n)+log(2)/2 \\ Charles R Greathouse IV, Feb 20 2012

sumalt(k=2, -(-1)^k * zeta'(k)) \\ Vaclav Kotesovec, Jun 17 2021

import mpmath
mpmath.mp.pretty=True; mpmath.mp.dps=108 #precision
mpmath.nsum(lambda n: (-1+mpmath.zeta(1+n))/n, [1,mpmath.inf]) # Peter Luschny, Jul 14 2012

numerical_approx(sum((zeta(k+1)-1)/k for k in [1..1000]), digits=120) # G. C. Greubel, Nov 15 2018

Equals Sum_{n>=2} log(n/(n-1))/n = Sum_{n>=1, k>=2} 1/(n*k^(n+1)). [From Mathworld links]
Equals -Sum_{k>=2} (-1)^k * zeta'(k). - Vaclav Kotesovec, Jun 17 2021
Equals log(A245254) = Sum_{k>=1} log(k)/(k*(k+1)). - Amiram Eldar, Jun 27 2021
Equals -log(A242624). - Amiram Eldar, Feb 06 2022

A124175 Decimal expansion of Product_{primes p} ((p-1)/p)^(1/p).

Original entry on oeis.org

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

Offset: 0

David W. Wilson, Dec 05 2006

This might be interpreted as the expected value of phi(n)/n for very large n.

			0.5598656169323734857237622442234167172576663702129060395542339339\
352031717975915936276540950630665470795373094197373037280781542375...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 164.
Mathoverflow, Asymptotics of product of Euler's totient function, 2016.
Eric Weisstein's World of Mathematics, Prime Zeta Function

Cf. A126226, A085548, A085541, A085964 - A085969, A242624, A272028.

digits = 100; s = Exp[-NSum[PrimeZetaP[h+1]/h, {h, 1, Infinity}, WorkingPrecision -> digits+5, NSumTerms -> 3 digits]]; RealDigits[s, 10, digits][[1]] (* Jean-François Alcover, Dec 07 2015, after Robert Gerbicz *)

default(realprecision,256);(f(k)=return(sum(n=1,512,moebius(n)/n*log(zeta(k*n)))));exp(sum(h=1,512,-1/h*f(h+1))) /* Robert Gerbicz */

exp(-suminf(m=2,log(zeta(m))*sumdiv(m,k,if(kMartin Fuller */

exp(-suminf(h=1, primezeta(h+1)/h)). - Robert Gerbicz
[Notation not clear. Is this perhaps exp(-Sum_{h=1..oo} primezeta(h+1)/h) ? - N. J. A. Sloane, Oct 08 2017]
Equals exp(1) * lim_{n->infinity} (A001088(n)/n!)^(1/n). - Vaclav Kotesovec, Feb 05 2016

Robert Gerbicz computed this to 130 decimal places.

A245254 Decimal expansion of U = Product_{k>=1} (k^(1/(k*(k+1)))), a Khintchine-like limiting constant related to Lüroth's representation of real numbers.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jul 15 2014

The geometric mean of the Yule-Simon distribution with parameter value 1 (A383855) approaches this constant. In general, the geometric mean of the Yule-Simon distribution approaches Product_{k>=2} k^(1/(p*Beta(k,p+1))). - Jwalin Bhatt, May 12 2025

			2.200161058099026553194557866559944872685662324752723888723145117763169...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.8.1 Alternative representations [of real numbers], p. 62.

Sofia Kalpazidou, Khintchine's constant for Lüroth representation, Journal of Number Theory, Vol. 29, No. 2 (June 1988), pp. 196-205.

Cf. A002210, A085291, A085361, A242624, A244109(V), A131688(W), A383855.

evalf(exp(Sum((Zeta(n+1)-1)/n, n=1..infinity)), 120); # Vaclav Kotesovec, Dec 11 2015

Exp[NSum[Log[k]/(k*(k+1)), {k, 1, Infinity}, WorkingPrecision -> 120, NSumTerms -> 5000, Method -> {NIntegrate, MaxRecursion -> 100}]] (* Vaclav Kotesovec, Dec 11 2015 *)

Equals exp(A085361).
U*V*W = 1, where V is A244109 and W is A131688.
Equals e * A085291. - Amiram Eldar, Jun 27 2021
Equals 1/A242624. - Amiram Eldar, Feb 06 2022

Corrected by Vaclav Kotesovec, Dec 11 2015

A242623 Decimal expansion of Product_{n>1} (1+1/n)^(1/n).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, May 19 2014

			1.758743627951184824699896849661932...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.9 p. 122.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A131688, A242624, A244625.

SetDefaultRealField(RealField(100)); L:=RiemannZeta(); Exp((&+[(-1)^n*(Evaluate(L,n)-1)/(n-1): n in [2..10^3]])); // G. C. Greubel, Nov 15 2018

evalf(exp(sum((-1)^(n+1)*Zeta(n+1)/n, n=1..infinity))/2, 120); # Vaclav Kotesovec, Dec 11 2015

Exp[NSum[((-1)^n*(-1 + Zeta[n]))/(n - 1), {n, 2, Infinity}, NSumTerms -> 300, WorkingPrecision -> 105] ] // RealDigits[#, 10, 103]& // First (* edited by Jean-François Alcover, May 23 2014 *)

default(realprecision, 100); exp(suminf(n=2, (-1)^n*(zeta(n)-1)/(n-1))) \\ G. C. Greubel, Nov 15 2018

numerical_approx(exp(sum((-1)^k*(zeta(k)-1)/(k-1) for k in [2..1000])), digits=100) # G. C. Greubel, Nov 15 2018

Equals exp(A131688)/2.

Data extended by Jean-François Alcover, May 23 2014

A244625 Decimal expansion of Product_{n>1} (1 - 1/n^2)^(1/n).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jul 02 2014

			0.7993704013063328789872528539753525668777...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.9 Alladi-Grinstead Constant, p. 122.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A085361, A242623, A242624.

SetDefaultRealField(RealField(100)); L:=RiemannZeta();  Exp(-(&+[(Evaluate(L, 2*n+1)-1)/n: n in [1..10^3]])); // G. C. Greubel, Nov 15 2018

evalf(exp(-sum((Zeta(2*n+1)-1)/n, n=1..infinity)), 120); # Vaclav Kotesovec, Dec 11 2015

digits = 102; Exp[-NSum[(Zeta[2*n+1]-1)/n, {n, 1, Infinity}, NSumTerms -> 300, WorkingPrecision -> digits+10]] // RealDigits[#, 10, digits]& // First

default(realprecision, 100); exp(-suminf(n=1, (zeta(2*n+1)-1)/n)) \\ G. C. Greubel, Nov 15 2018

numerical_approx(exp(-sum((zeta(2*n+1)-1)/n for n in [1..1000])), digits=100) # G. C. Greubel, Nov 15 2018

Equals exp(-Sum_{n>0} (zeta(2*n+1) - 1)/n).
Equals A242623 * A242624.
Also equals A242623 * exp(-A085361).

cp's OEIS Frontend

A085361 Decimal expansion of the number c = Sum_{n>=1} (zeta(n+1)-1)/n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Sage

Sage

Formula

A124175 Decimal expansion of Product_{primes p} ((p-1)/p)^(1/p).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A245254 Decimal expansion of U = Product_{k>=1} (k^(1/(k*(k+1)))), a Khintchine-like limiting constant related to Lüroth's representation of real numbers.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A242623 Decimal expansion of Product_{n>1} (1+1/n)^(1/n).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

SageMath

Formula

Extensions

A244625 Decimal expansion of Product_{n>1} (1 - 1/n^2)^(1/n).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula