A175475 -id:A175475 - cp's OEIS frontend

A090081 Cube root-smooth numbers: numbers k whose largest prime factor does not exceed the cube root of k.

1, 8, 16, 27, 32, 36, 48, 54, 64, 72, 81, 96, 108, 125, 128, 135, 144, 150, 160, 162, 180, 192, 200, 216, 225, 240, 243, 250, 256, 270, 288, 300, 320, 324, 343, 350, 360, 375, 378, 384, 392, 400, 405, 420, 432, 441, 448, 450, 480, 486, 490, 500, 504, 512, 525

Offset: 1

Labos Elemer, Nov 21 2003

What is the asymptotic growth of this sequence?

Answer: a(n) ~ k*n, where k = 1/A175475. That is, about 4.8% of numbers are in this sequence. - Charles R Greathouse IV, Jul 14 2014

			378 = 2 * 3^3 * 7 is a term of the sequence since 7 < 7.23... = 378^(1/3).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A063763, A001248, A048098, A036966, A054744, A059172, A068936, A076405, A076467, A006530, A175475.

filter:= n ->
evalb(max(seq(f[1],f=ifactors(n)[2]))^3 <= n):
select(filter, [$1..1000]); # Robert Israel, Jul 14 2014

ffi[x_] := Flatten[FactorInteger[x]]; ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}]; lf[x_] := Length[FactorInteger[x]]; ma[x_] := Max[ba[x]]; Do[If[ !Greater[ma[n], gy=n^(1/3)//N]&&!PrimeQ[n], Print[n(*, {gy, ma[n]}*)]], {n, 1, 1000}]
Select[Range[1000], (FactorInteger[#][[-1,1]])^3 <= # &] (* T. D. Noe, Sep 14 2011 *)
Select[Range[1000],FactorInteger[#][[-1,1]]<=CubeRoot[#]&] (* Harvey P. Dale, Jun 30 2025 *)

is(n)=my(f=factor(n)[,1]);f[#f]^3<=n \\ Charles R Greathouse IV, Sep 14 2011

from sympy import primefactors
def ok(n):
    if n==1 or max(primefactors(n))**3<=n: return True
    else: return False
print([n for n in range(1, 1001) if ok(n)]) # Indranil Ghosh, Apr 23 2017

Solutions to A006530(n) <= n^(1/3).

A245238 Decimal expansion of the Dickman function evaluated at 1/4.

Original entry on oeis.org

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

Offset: 0

Charles R Greathouse IV, Jul 14 2014

Density of the fourth-root-smooth numbers.

			F(1/4) = 0.00491092564776083235273915092361518603248429741769294597796...

Karl Dickman, On the frequency of numbers containing prime factors of a certain relative magnitude, Arkiv för Matematik, Astronomi och Fysik 22A 10 (1930), pp. 1-14.

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

F(1/2) = A244009, F(1/3) = A175475.

RealDigits[1-Log[4]+PolyLog[2, 1/4]+2*Log[2]^2-Pi^2/12-PolyLog[3, 1/4]-PolyLog[2, 1/4]*Log[2]-2/3*Log[2]^3+13*Zeta[3]/24,10,100,-1][[1]] (* Vaclav Kotesovec, Jul 15 2014 *)

1-log(4)+polylog(2,1/4)+2*log(2)^2-Pi^2/12-polylog(3,1/4)-polylog(2,1/4)*log(2)-2/3*log(2)^3+13*zeta(3)/24

A258945 Decimal expansion of Dickman's constant C_4.

Original entry on oeis.org

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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A090081 Cube root-smooth numbers: numbers k whose largest prime factor does not exceed the cube root of k.

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A245238 Decimal expansion of the Dickman function evaluated at 1/4.

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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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