A245238 -id:A245238 - cp's OEIS frontend

A258945 Decimal expansion of Dickman's constant C_4.

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

Offset: 0

Jean-François Alcover, Jun 15 2015

			0.067645202106946136969750231033822993923421934494920431730186...

David Broadhurst, Dickman polylogarithms and their constants arXiv:1004.0519 [math-ph], 2010.

Cf. A175475, A245238.

digits = 103; C4 = NIntegrate[(Log[x/(2*x+1)]*PolyLog[2, x] + (1/2)*Log[x]^2* PolyLog[1, -2*x])/(x*(x+1)), {x, 0, 1/2}, WorkingPrecision -> digits+5] + 3*PolyLog[4, 1/2] - 3/8 *PolyLog[4, 1/4] - 3/4* Log[2]*PolyLog[3, 1/4] + (Pi^2 - 9*Log[2]^2)/12*PolyLog[2, 1/4] + 21*Log[2]*Zeta[3]/8 + Pi^2*(Log[2]^2/24) - Pi^2*Log[2]*(Log[3]/6) + Log[2]^3*Log[3]/2 - 5*Log[2]^4/8; Join[{0}, RealDigits[C4, 10, digits] // First]

from mpmath import mp, log, polylog, zeta, pi, quad
mp.dps=104
f=lambda x: (log(x/(2*x+1))*polylog(2, x) + (1/2)*log(x)**2*polylog(1, -2*x))/(x*(x+1))
I=quad(f, [0, 1/2]) + 3*polylog(4, 1/2) - 3/8*polylog(4, 1/4) - 3/4*log(2) * polylog(3, 1/4) +(pi**2 - 9*log(2)**2)/12*polylog(2, 1/4) + 21*log(2)*zeta(3)/8 + pi**2*(log(2)**2/24) - pi**2*log(2)*(log(3)/6) + log(2)**3*log(3)/2 - 5*log(2)**4/8
print([int(z) for z in list(str(I)[2:-1])]) # Indranil Ghosh, Jul 03 2017

C_1 = 0, C_2 = -Pi^2/12, C_3 = -zeta(3)/3.
C_4 = Integral_{0..1/2} (log(x/(2*x+1))*polylog(2, x) + (1/2)*log(x)^2*polylog(1, -2*x))/(x*(x+1)) dx + 3*polylog(4, 1/2) - 3/8*polylog(4, 1/4) - 3/4*log(2) * polylog(3, 1/4) +(Pi^2 - 9*log(2)^2)/12*polylog(2, 1/4) + 21*log(2)*zeta(3)/8 + Pi^2*(log(2)^2/24) - Pi^2*log(2)*(log(3)/6) + log(2)^3*log(3)/2 - 5*log(2)^4/8.
Also (conjecturally) equals Pi^4/1440.

A309638 Nearest integer to 1/F(1/x), where F(x) is the Dickman function.

Original entry on oeis.org

1, 3, 21, 204, 2819, 50891, 1143423, 30939931, 984011503, 36098843631, 1504934136432, 70436763188525, 3664092112471681, 210056231435360023, 13175390260774094846, 898537704166507324228, 66265550246147429710863, 5259409287834480235626661, 447341910388133084658686126, 40620967386538406952534036284

Offset: 1

Jeremy Tan, Aug 11 2019

nonn

The asymptotic density of the n-th-root-smooth numbers is approximately 1/a(n).

Van de Lune and Wattel show a(n) >= A001147(n) for n >= 1.

			The asymptotic density of fifth-root-smooth numbers is F(1/5) = 0.000354724700... = 1/2819.08758..., so a(5) = 2819.

G. Marsaglia, A. Zaman and J. Marsaglia (1989), Numerical Solution of Some Classical Differential-Difference Equations, Mathematics of Computation, 53 (187), 191-201.
Jeremy Tan, Python program
J. van de Lune and E. Wattel (1969), On the Numerical Solution of a Differential-Difference Equation Arising in Analytic Number Theory, Mathematics of Computation, 23 (106), 417-421.
Eric Weisstein's World of Mathematics, Dickman Function

F(1/2) = A244009; F(1/3) = A175475; F(1/4) = A245238.

1/F(1/x) = 1/rho(x), where rho(x) satisfies rho'(x) = -rho(x-1)/x and rho(x) = 1 for x <= 1. rho(x) may be computed to arbitrary precision by the method of Marsaglia, Zaman and Marsaglia (implemented in the Python program in Links).

a(n) ~ exp(Ei(t) - n*t) / (t * sqrt(2*Pi*n)), where Ei is the exponential integral and t is the positive root of exp(t) - n*t - 1 (van de Lune and Wattel).

A344475 Decimal expansion of the value of the Dickman function at phi + 1 = phi^2 = (3 + sqrt(5))/2 (A104457).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, May 20 2021

			0.10464776377316485385416972771819339482414269115729...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 286.

Pieter Moree, A special value of Dickman's function, Math. Student, Vol. 64 (1995), pp. 47-50; entire volume.
Pieter Moree, Nicolaas Govert de Bruijn, the enchanter of friable integers, Indagationes Mathematicae, Vol. 24, No. 4 (2013), pp. 774-801.
Eric Weisstein's World of Mathematics, Dickman Function.
Wikipedia, Dickman function.

Cf. A000796, A001622, A002390, A013661, A104457, A344476.

Cf. A175475, A244009 (value at 1/2), A245238, A309638.

RealDigits[1 - 2*Log[GoldenRatio] + Log[GoldenRatio]^2 - Pi^2/60, 10, 100][[1]]

my(phi = quadgen(5)); 1 - 2*log(phi) + log(phi)^2 - Pi^2/60 \\ Amiram Eldar, Jan 09 2025

Equals 1 - 2*log(phi) + log(phi)^2 - Pi^2/60 (Moree, 1995).

More terms from Amiram Eldar, Jan 09 2025

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A258945 Decimal expansion of Dickman's constant C_4.

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A309638 Nearest integer to 1/F(1/x), where F(x) is the Dickman function.

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A344475 Decimal expansion of the value of the Dickman function at phi + 1 = phi^2 = (3 + sqrt(5))/2 (A104457).

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