A245254 -id:A245254 - 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

A242624 Decimal expansion of Product_{n>1} (1-1/n)^(1/n).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, May 19 2014

			0.4545121805146463170328014636843273993...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.9, pp. 121-122.

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

Cf. A085361, A242623, A244625, A245254.

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

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

Exp[-NSum[(1-Zeta[n])/(1-n), {n, 2, Infinity}, NSumTerms -> 300, WorkingPrecision -> 110]] // RealDigits[#, 10, 100]& // First

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

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

From Amiram Eldar, Feb 06 2022: (Start)
Equals exp(-A085361).
Equals 1/A245254. (End)

Mma modified and data extended by Jean-François Alcover, May 23 2014

A383855 The n-th term of the sequence is k after every k*(k+1)/2 occurrences of 1, with multiple values following a 1 listed in order.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 2, 1, 4, 1, 1, 2, 3, 1, 1, 1, 2, 5, 1, 1, 1, 2, 3, 1, 1, 4, 1, 2, 6, 1, 1, 1, 2, 3, 1, 1, 1, 2, 1, 7, 1, 1, 2, 3, 4, 5, 1, 1, 1, 2, 1, 1, 1, 2, 3, 8, 1, 1, 1, 2, 1, 4, 1, 1, 2, 3, 6, 1, 1, 1, 2, 5, 9, 1, 1, 1, 2, 3, 1, 1, 4, 1, 2, 1, 1, 1, 2, 3, 1, 10

Offset: 1

Jwalin Bhatt, May 12 2025

nonn

The frequencies of the terms follow the Yule-Simon distribution with parameter value 1. The geometric mean approaches A245254 in the limit.

			After every ((2*3)/2=3) ones we see a 2,
after every ((3*4)/2=6) ones we see a 3,
after every ((4*5)/2=10) ones we see a 4 and so on.

Jwalin Bhatt, Table of n, a(n) for n = 1..10000
Wikipedia, Yule-Simon distribution

Cf. A245254, A383899.

from itertools import islice
def beta_distribution_generator():
    num_ones, num_reached = 0, 1
    while num_ones := num_ones+1:
        yield 1
        for num in range(2, num_reached+2):
            if num_ones % (num*(num+1)//2) == 0:
                yield num
                num_reached += num == num_reached+1
A383855 = list(islice(beta_distribution_generator(), 120))

A383899 A sequence constructed by greedily sampling the Yule-Simon distribution for parameter value 1, to minimize discrepancy selecting the smallest value in case of ties.

Original entry on oeis.org

1, 2, 1, 3, 1, 4, 1, 2, 1, 5, 1, 6, 1, 2, 1, 3, 1, 7, 1, 2, 1, 8, 1, 9, 1, 2, 1, 4, 1, 3, 1, 2, 1, 10, 1, 11, 1, 2, 1, 3, 1, 5, 1, 2, 1, 4, 1, 12, 1, 2, 1, 3, 1, 13, 1, 2, 1, 6, 1, 14, 1, 2, 1, 3, 1, 4, 1, 2, 1, 5, 1, 15, 1, 2, 1, 7, 1, 3, 1, 2, 1, 16, 1, 17

Offset: 1

Jwalin Bhatt, May 14 2025

nonn

The geometric mean approaches A245254 in the limit.

The probability mass function of the Yule-Simon distribution with parameter 1 is given by p(k) = 1/(k*(k+1)) for k >= 1.

			Let p(k) denote the probability of k and c(k) denote the number of occurrences of k among the first n-1 terms; then the expected number of occurrences of k among n random terms is given by n*p(k).
We subtract the actual occurrences c(k) from the expected occurrences and pick the one with the highest value.
| n | n*p(1) - c(1) | n*p(2) - c(2) | n*p(3) - c(3) | choice |
|---|---------------|---------------|---------------|--------|
| 1 |     0.5       |     0.166     |     0.083     |   1    |
| 2 |     0         |     0.333     |     0.166     |   2    |
| 3 |     0.5       |    -0.5       |     0.25      |   1    |
| 4 |     0         |    -0.333     |     0.333     |   3    |
| 5 |     0.5       |    -0.166     |    -0.583     |   1    |

Jwalin Bhatt, Table of n, a(n) for n = 1..10000
Wikipedia, Yule-Simon distribution

Cf. A245254, A383855.

probCountDiff[j_, k_, count_]:=k/(j*(j+1))-Lookup[count, j, 0]
samplePDF[n_]:=Module[{coeffs, unreachedVal, counts, k, probCountDiffs, mostProbable},
  coeffs=ConstantArray[0, n]; unreachedVal=1; counts=<||>;
  Do[probCountDiffs=Table[probCountDiff[i, k, counts], {i, 1, unreachedVal}];
    mostProbable=First@FirstPosition[probCountDiffs, Max[probCountDiffs]];
    If[mostProbable==unreachedVal, unreachedVal++]; coeffs[[k]]=mostProbable;
    counts[mostProbable]=Lookup[counts, mostProbable, 0]+1; , {k, 1, n}]; coeffs]
A383899=samplePDF[120]

A272286 Decimal expansion of Product_{k >= 1} (k(k+1))^(-1/(k(k+1))), a constant related to the alternating Lüroth representations of real numbers.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 24 2016

			0.1292150184060998413415719000742197771573364620386787448773...

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, Volume 29, Issue 2, June 1988, Pages 196-205.

Cf. A131688, A244109, A245254.

digits = 105; Exp[-NSum[((1 + (-1)^(n + 1))*Zeta[n + 1] - 1)/n, {n, 1, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> 2 digits, NSumTerms -> 200]] // RealDigits[#, 10, digits]& // First

Exp(-Sum_{n >= 1} (((1 + (-1)^(n+1))*Zeta(n+1) - 1)/n)). - After Vaclav Kotesovec's formula for A244109.

Offset corrected by Andrey Zabolotskiy, Dec 12 2023

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

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A242624 Decimal expansion of Product_{n>1} (1-1/n)^(1/n).

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A383855 The n-th term of the sequence is k after every k*(k+1)/2 occurrences of 1, with multiple values following a 1 listed in order.

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A383899 A sequence constructed by greedily sampling the Yule-Simon distribution for parameter value 1, to minimize discrepancy selecting the smallest value in case of ties.

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Mathematica

A272286 Decimal expansion of Product_{k >= 1} (k*(k+1))^(-1/(k*(k+1))), a constant related to the alternating Lüroth representations of real numbers.

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Author

Keywords

Examples

References

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Crossrefs

Programs

Mathematica

Formula

Extensions

A272286 Decimal expansion of Product_{k >= 1} (k(k+1))^(-1/(k(k+1))), a constant related to the alternating Lüroth representations of real numbers.