A217598 -id:A217598 - cp's OEIS frontend

A247667 Decimal expansion of the coefficient c_m in c_m*log(N), the asymptotic mean number of factors in a random factorization of n <= N.

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

Offset: 0

Jean-François Alcover, Sep 22 2014

			0.55001000541315449183305812670222219646116822710271404...

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. 41.

Cf. A107311, A129373, A129374, A129375, A217598, A247668.

digits = 101; rho = x /. FindRoot[Zeta[x] == 2, {x, 2}, WorkingPrecision -> digits+5]; cm = -1/Zeta'[rho]; RealDigits[cm, 10, digits] // First

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

Residue_{s = rho} 1/(2 - Zeta(s)). - Vaclav Kotesovec, Nov 04 2018

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.

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A247667 Decimal expansion of the coefficient c_m in c_m*log(N), the asymptotic mean number of factors in a random factorization of n <= N.

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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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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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