A247667 -id:A247667 - cp's OEIS frontend

A107311 Decimal expansion of the solution to zeta(x) = 2.

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

Offset: 1

Ralf Stephan, May 20 2005

From Artur Jasinski, Dec 21 2024: (Start)
Borwein et al. (2007) proved (Theorem 3.1) that the real parts of the zeros of the partials sums of the Riemman zeta functions are not greater than this constant.
Conjecture 1: the real parts of the zeros of the prime zeta function are not greater than this constant.
Conjecture 2: the real parts of the zeros of the anyone subset of the prime zeta function are not greater than this constant. (End)

			zeta(1.72864723899818361813510301...) = 2.

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.5 Kalmar's Composition Constant, p. 293.

Peter Borwein, Greg Fee, Ron Ferguson, and Alexa van der Waal, Zeros of Partial Sums of the Riemann Zeta Function, Experiment. Math. 16(1) (2007), pp. 21-40. See p. 25.
Einar Hille, A problem in "factorisatio numerorum", Acta Arithmetica, 2(1);134-144, 1936.
Hsien-Kuei Hwang, Distribution of the number of factors in random ordered factorizations of integers, Journal of Number Theory 81:1 (2000), pp. 61-92.
M. Klazar and F. Luca, On the maximal order of numbers in the "factorisatio numerorum" problem, arXiv:math/0505352 [math.NT], 2005-2006.

Cf. A129374, A247667.

x /. FindRoot[ Zeta[x] == 2, {x, 2}, WorkingPrecision -> 102] // RealDigits // First (* Jean-François Alcover, Mar 19 2013 *)

PARI
```
solve(X=1.5,2,zeta(X)-2)
```

A129374 G.f. satisfies: A(x) = 1/(1-x) * A(x^2)A(x^3)A(x^4)...A(x^n)*...

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 15, 20, 35, 48, 76, 103, 166, 221, 333, 451, 671, 894, 1303, 1730, 2479, 3288, 4615, 6086, 8502, 11142, 15299, 20034, 27285, 35514, 47937, 62168, 83259, 107650, 142929, 184090, 243207, 312041, 409210, 523709, 683261, 871239, 1130703

Offset: 0

Paul D. Hanna, Apr 12 2007

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A074206, A129373, A129375.

{a(n)=local(A=1+x);for(i=2,n,A=1/(1-x)*prod(n=2,i,subst(A,x,x^n+x*O(x^i)))); polcoeff(A,n)}

G.f.: A(x) = Product_{n>=1} 1/(1 - x^n)^A074206(n) where A074206(n) equals the number of ordered factorizations of n.

a(n) ~ exp((1 + 1/r) * (-Gamma(1+r) * Zeta(1+r) / Zeta'(r))^(1/(1+r)) * n^(r/(1+r))) * (-Gamma(1+r) * Zeta(1+r) / Zeta'(r))^(1/(10*(1+r))) / ((2*Pi)^(29/50) * sqrt(1+r) * n^((6 + 5*r)/(10*(1+r)))), where r = A107311 = 1.7286472389981836181351... is the root of the equation Zeta(r) = 2, Zeta'(r) = -1/A247667. - Vaclav Kotesovec, Nov 04 2018

A129373 G.f. satisfies: A(x) = (1+x) * A(x^2)A(x^3)A(x^4)...A(x^n)*...

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 7, 9, 13, 19, 26, 34, 52, 67, 89, 123, 166, 214, 295, 380, 501, 660, 858, 1098, 1461, 1858, 2384, 3072, 3940, 4975, 6410, 8070, 10234, 12946, 16322, 20412, 25848, 32201, 40261, 50287, 62728, 77681, 96885, 119673, 148197, 183108, 225974

Offset: 0

Paul D. Hanna, Apr 12 2007

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A074206; A129374, A129375.

{a(n)=local(A=1+x);for(i=2,n,A=(1+x)*prod(n=2,i,subst(A,x,x^n+x*O(x^i)))); polcoeff(A,n)}

G.f.: A(x) = Product_{n>=1} (1 + x^n)^A074206(n) where A074206(n) equals the number of ordered factorizations of n.

a(n) ~ exp((1 + 1/r) * (-(1 - 2^(-r)) * Gamma(1+r) * Zeta(1+r) / Zeta'(r))^(1/(1+r)) * n^(r/(1+r))) * (-(1 - 2^(-r)) * Gamma(1+r) * Zeta(1+r) / Zeta'(r))^(1/(2 + 2*r)) / (2^(1/10) * sqrt(Pi) * sqrt(1+r) * n^((2+r)/(2 + 2*r))), where r = A107311 = 1.7286472389981836181351... is the root of the equation Zeta(r) = 2, Zeta'(r) = -1/A247667. - Vaclav Kotesovec, Nov 04 2018

A129375 E.g.f. satisfies: A(x) = exp(x) * A(x^2)A(x^3)A(x^4)...A(x^n)*...

Original entry on oeis.org

1, 1, 3, 13, 97, 621, 6571, 58633, 779073, 9317017, 138628531, 1977676581, 37384244833, 620735382853, 12434855135067, 245117537189281, 5651550278494081, 123266430844431153, 3128700944169196003

Offset: 0

Paul D. Hanna, Apr 12 2007

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..260

Cf. A074206, A129373, A129374.

{a(n)=local(A=1+x);for(i=2,n,A=exp(x+x*O(x^n))*prod(n=2,i,subst(A,x,x^n+x*O(x^i)))); n!*polcoeff(A,n)}

E.g.f.: A(x) = exp( Sum_{n>=1} A074206(n)*x^n ) where A074206(n) equals the number of ordered factorizations of n.

a(n) ~ n! * exp(2/5 + (1 + 1/r) * (-Gamma(1+r) / Zeta'(r))^(1/(1+r)) * n^(r/(1+r))) * (-Gamma(1+r) / Zeta'(r))^(1/(2*(1+r))) / (sqrt(2*Pi*(1+r)) * n^((2+r)/(2*(1+r)))), where r = A107311 = 1.7286472389981836181351... is the root of the equation Zeta(r) = 2, Zeta'(r) = -1/A247667. - Vaclav Kotesovec, Nov 04 2018

PARI program fixed by Vaclav Kotesovec, Feb 26 2014

A318767 G.f. satisfies: A(x) = (1+x)/(1-x) * A(x^2)A(x^3)A(x^4)...A(x^n)*... .

Original entry on oeis.org

1, 2, 4, 8, 16, 28, 52, 88, 152, 252, 416, 664, 1076, 1684, 2636, 4060, 6248, 9444, 14292, 21312, 31748, 46796, 68804, 100200, 145784, 210240, 302520, 432428, 616716, 873972, 1236136, 1738560, 2439936, 3407924, 4749160, 6589156, 9123976, 12582620, 17316052, 23745756

Offset: 0

Seiichi Manyama, Nov 04 2018

nonn

Convolution of A129373 and A129374. - Vaclav Kotesovec, Nov 05 2018

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A074206, A129373, A129374.

G.f.: Product_{k>=1} ((1 + x^k)/(1 - x^k))^A074206(k) where A074206(n) is the number of ordered factorizations of n.

a(n) ~ exp((1+r) * ((2^(1+r) - 1) * Gamma(1+r) * Zeta(1+r))^(1/(1+r)) * n^(r/(1+r)) / (r * 2^(r/(1+r)) * (-Zeta'(r))^(1/(1+r)))) * (-2*(2^(1+r) - 1) * Gamma(1+r) * Zeta(1+r) / Zeta'(r))^(1/(10*(1+r))) / (2^(7/25) * Pi^(29/50) * sqrt(1+r) * n^((6+5*r)/(10*(1+r)))), where r = A107311 = 1.7286472389981836181351... is the root of the equation Zeta(r) = 2, Zeta'(r) = -1/A247667. - Vaclav Kotesovec, Nov 05 2018

A307615 E.g.f. A(x) satisfies: A(x) = exp(x) * A(x^2)^2A(x^3)^3A(x^4)^4* ... A(x^k)^k ...

Original entry on oeis.org

1, 1, 5, 31, 337, 2741, 40621, 474475, 8461601, 132034537, 2648537461, 50079699671, 1204884343345, 26450428964701, 697107087763997, 17873985363570211, 526080367468142401, 15060611189639187665, 487251625325328212581, 15494976568071805188367, 545902629556769672596241

Offset: 0

Ilya Gutkovskiy, Apr 18 2019

nonn

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 31*x^3/3! + 337*x^4/4! + 2741*x^5/5! + 40621*x^6/6! + 474475*x^7/7! + 8461601*x^8/8! + 132034537*x^9/9! + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..420

Cf. A050369, A074206, A129375, A307604.

terms = 20; A[] = 1; Do[A[x] = Exp[x] Product[A[x^k]^k, {k, 2, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x] Range[0, terms]!

E.g.f.: exp(Sum_{k>=1} A050369(k)*x^k).
a(0) = 1; a(n) = Sum_{k=1..n} A074206(k)*k*k!*binomial(n-1,k-1)*a(n-k).
a(n) ~ (-Gamma(2+r)/zeta'(r))^(1/(4 + 2*r)) * exp(-n + 12/25 + n^(1 - 1/(2+r)) * (2+r) * (-Gamma(2+r)/zeta'(r))^(1/(2+r)) / (1+r)) * n^(n - 1/(4 + 2*r)) / sqrt(2+r), where r = A107311 = 1.7286472389981836181351... is the root of the equation Zeta(r) = 2, Zeta'(r) = -1/A247667. - Vaclav Kotesovec, Aug 09 2021

A247668 Decimal expansion of the coefficient c_v in c_v*log(N), the asymptotic variance of the number of factors in a random factorization of n <= N.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Sep 22 2014

			0.308403444608077001633607726174587986672094960536886...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.5. Kalmár’s Composition Constant, p. 293.

Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 37.

Cf. A107311, A217598, A247667.

digits = 102; rho = x /. FindRoot[Zeta[x] == 2, {x, 2}, WorkingPrecision -> digits+5]; cv = (-1/Zeta'[rho])*(Zeta''[rho]/Zeta'[rho]^2 - 1); RealDigits[cv, 10, digits] // First

c_v = (-1/zeta'(rho))*(zeta''(rho)/zeta'(rho)^2 - 1), where rho = 1.728647... is A107311, the real solution to zeta(rho) = 2.

A329110 Number of integer sequences 1 <= b_1 < b_2 < ... < b_t <= n such that b_i divides b_(i+1) for all 0 < i < t.

Original entry on oeis.org

1, 3, 5, 9, 11, 17, 19, 27, 31, 37, 39, 55, 57, 63, 69, 85, 87, 103, 105, 121, 127, 133, 135, 175, 179, 185, 193, 209, 211, 237, 239, 271, 277, 283, 289, 341, 343, 349, 355, 395, 397, 423, 425, 441, 457, 463, 465, 561, 565, 581, 587, 603, 605, 645, 651, 691

Offset: 1

Peter Kagey, Nov 04 2019

nonn

Cumulative sum of A067824.

			For n = 4 the a(4) = 9 sequences are 1; 1, 2; 1, 2, 4; 1, 3; 1, 4; 2; 2, 4; 3; and 4.

Peter Kagey, Table of n, a(n) for n = 1..10000
Mathematics Stack Exchange user Markus Scheuer, Finding the number of subsets of a set such that an element divides the succeeding element.

Cf. A067824.

s=0; for (n=1, #(z=vector(56)), print1 (s += z[n]=1+sumdiv(n, k, if (kRémy Sigrist, Nov 08 2019

From Vaclav Kotesovec, Mar 18 2021: (Start)
a(n) ~ -2*n^r/(r*zeta'(r)), where r=A107311 is the root of the equation zeta(r)=2.
a(n) ~ 2*A247667 * n^A107311 / A107311.
a(n) ~ 2*A217598 * n^A107311. (End)

A247605 Decimal expansion of the coefficient c_md in c_md*log(N)^(1/rho), the asymptotic mean number of distinct factors in a random factorization of n <= N.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Sep 22 2014

			1.48791597167815789287168630546556607279198849...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.5. Kalmár’s Composition Constant, p. 293.

Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2024, p. 37.

Cf. A107311, A217598, A247667.

digits = 101; rho = x /. FindRoot[Zeta[x] == 2, {x, 2}, WorkingPrecision -> digits + 5]; cmd = (-1/rho)*Gamma[-1/rho]*(-1/Zeta'[rho])^(1/rho); RealDigits[cmd, 10, digits] // First

c_md = (-1/rho)*Gamma(-1/rho)*(-1/zeta'(rho))^(1/rho), where rho = 1.728647... is A107311, the real solution to zeta(rho) = 2.

cp's OEIS Frontend

A107311 Decimal expansion of the solution to zeta(x) = 2.

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A129374 G.f. satisfies: A(x) = 1/(1-x) * A(x^2)*A(x^3)*A(x^4)*...*A(x^n)*...

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A129373 G.f. satisfies: A(x) = (1+x) * A(x^2)*A(x^3)*A(x^4)*...*A(x^n)*...

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A129375 E.g.f. satisfies: A(x) = exp(x) * A(x^2)*A(x^3)*A(x^4)*...*A(x^n)*...

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A318767 G.f. satisfies: A(x) = (1+x)/(1-x) * A(x^2)*A(x^3)*A(x^4)*...*A(x^n)*... .

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A307615 E.g.f. A(x) satisfies: A(x) = exp(x) * A(x^2)^2*A(x^3)^3*A(x^4)^4* ... *A(x^k)^k* ...

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A247668 Decimal expansion of the coefficient c_v in c_v*log(N), the asymptotic variance of the number of factors in a random factorization of n <= N.

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A329110 Number of integer sequences 1 <= b_1 < b_2 < ... < b_t <= n such that b_i divides b_(i+1) for all 0 < i < t.

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A129374 G.f. satisfies: A(x) = 1/(1-x) * A(x^2)A(x^3)A(x^4)...A(x^n)*...

A129373 G.f. satisfies: A(x) = (1+x) * A(x^2)A(x^3)A(x^4)...A(x^n)*...

A129375 E.g.f. satisfies: A(x) = exp(x) * A(x^2)A(x^3)A(x^4)...A(x^n)*...

A318767 G.f. satisfies: A(x) = (1+x)/(1-x) * A(x^2)A(x^3)A(x^4)...A(x^n)*... .

A307615 E.g.f. A(x) satisfies: A(x) = exp(x) * A(x^2)^2A(x^3)^3A(x^4)^4* ... A(x^k)^k ...