A107311 -id:A107311 - cp's OEIS frontend

A307606 G.f. A(x) satisfies: A(x) = ((1 + x)/(1 - x)) * A(x^2)^2A(x^3)^3A(x^4)^4* ... A(x^k)^k ...

1, 2, 6, 16, 46, 104, 268, 596, 1406, 3060, 6812, 14356, 30948, 63660, 132328, 267164, 541678, 1072000, 2127052, 4140340, 8060588, 15458948, 29602504, 55990780, 105693252, 197422424, 367793952, 679206200, 1250557768, 2284986580, 4162202864, 7530956532, 13583095710

Offset: 0

Ilya Gutkovskiy, Apr 18 2019

nonn

Convolution of A307604 and A307605.

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 16*x^3 + 46*x^4 + 104*x^5 + 268*x^6 + 596*x^7 + 1406*x^8 + 3060*x^9 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A050369, A074206, A307604, A307605, A318767.

terms = 32; A[] = 1; Do[A[x] = (1 + x)/(1 - x) Product[A[x^k]^k, {k, 2, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]

G.f.: Product_{k>=1} ((1 + x^k)/(1 - x^k))^(k*A074206(k)).

a(n) ~ ((1 - 2^(2+r)) * Gamma(2+r) * zeta(2+r))^(1/(50*(2+r))) * exp(12/625 + 2^(1/(2+r) - 1) * (2+r) * ((1 - 2^(2+r)) * Gamma(2+r) * zeta(2+r))^(1/(2+r)) / (zeta'(r)^(1/(2+r)) * (1+r)) * n^((1+r)/(2+r))) / (A^(144/625) * 2^((3 + 2*r)/(50*(2 + r))) * zeta'(r)^(1/(50*(2+r))) * sqrt(Pi*(2+r)) * n^(1/2 + 1/(50*(2+r)))), where r = A107311 is the root of the equation zeta(r)=2 and A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Mar 18 2021

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)

A335494 Decimal expansion of s, where s is the root of Product_{k>=1} zeta(k*s) = 2.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Jun 11 2020

			1.8868691498777028184721160405750829854817368935543730821633756671766...

Eric Weisstein's World of Mathematics, Riemann Zeta Function.

Cf. A021002, A107311, A244285, A329385.

A342203 Decimal expansion of solution to zeta(x) = e.

Original entry on oeis.org

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

Offset: 1

Koksal Karakus, Mar 04 2021

			zeta(1.4744642873...) = e.

Cf. A001113 (e), A107311.

RealDigits[x /. FindRoot[Zeta[x] == E, {x, 2}, WorkingPrecision -> 105], 10, 100][[1]] (* Amiram Eldar, Mar 05 2021 *)

solve(x=1.1, 2, zeta(x)-exp(1)) \\ Michel Marcus, Mar 05 2021

More digits from Alois P. Heinz, Mar 05 2021

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.

A342220 Decimal expansion of solution to zeta(x) = Pi.

Original entry on oeis.org

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

Offset: 1

Koksal Karakus, Mar 05 2021

			zeta(1.39425321984488839...) = Pi.

Cf. A000796 (Pi), A107311, A342203.

RealDigits[x /. FindRoot[Zeta[x] - Pi, {x, 2}, WorkingPrecision -> 120], 10, 100][[1]] (* Amiram Eldar, May 31 2021 *)

solve(x=1.1, 2, zeta(x) - Pi) \\ Michel Marcus, Mar 07 2021

More terms from Jon E. Schoenfield, Mar 07 2021

A349604 Decimal expansion of the positive real solution to (1 - 1/2^x) * zeta(x) = 2.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Nov 23 2021

This constant, c, appears in the inequality A074206(n) <= n^c for odd n (Baustian and Bobkov, 2020).

			1.37778516983754118384089490370869137916464016638683...

Falko Baustian and Vladimir Bobkov, On asymptotic behavior of Dirichlet inverse, International Journal of Number Theory, Vol. 16, No. 6 (2020), pp. 1337-1354; arXiv preprint, arXiv:1903.12445 [math.NT], 2019-2020.

Cf. A074206, A107311.

RealDigits[s /. FindRoot[(1 - 1/2^s)*Zeta[s] == 2, {s, 2}, WorkingPrecision -> 110], 10, 100][[1]]

solve(x=1.1, 2, (1-1/2^x)*zeta(x) - 2) \\ Michel Marcus, Nov 23 2021

cp's OEIS Frontend

A307606 G.f. A(x) satisfies: A(x) = ((1 + x)/(1 - x)) * A(x^2)^2*A(x^3)^3*A(x^4)^4* ... *A(x^k)^k* ...

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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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A335494 Decimal expansion of s, where s is the root of Product_{k>=1} zeta(k*s) = 2.

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A342203 Decimal expansion of solution to zeta(x) = e.

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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.

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A307606 G.f. A(x) satisfies: A(x) = ((1 + x)/(1 - x)) * A(x^2)^2A(x^3)^3A(x^4)^4* ... A(x^k)^k ...

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 ...