A090081 -id:A090081 - cp's OEIS frontend

A175475 Decimal expansion of the Dickman function evaluated at 1/3.

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

Offset: 0

R. J. Mathar, May 25 2010

Density of the cube root-smooth numbers, see A090081. - Charles R Greathouse IV, Jul 14 2014

			F(1/3) = 0.04860838829113156690718...

David Broadhurst, Dickman polylogarithms and their constants arXiv:1004.0519 [math-ph], 2010.
K. Soundararajan, An asymptotic expansion related to the Dickman function, arXiv:1005.3494 [math.NT], 2010.

Cf. A002391, A072691, A175478.

N[1 - Log[3] + Log[3]^2/2 - Pi^2/12 + PolyLog[2, 1/3], 105] // RealDigits // First // Prepend[#, 0]& (* Jean-François Alcover, Feb 05 2013 *)

1-log(3)+log(3)^2/2-Pi^2/12+polylog(2,1/3) \\ Charles R Greathouse IV, Jul 14 2014

Equals 1 - log(3) + log^2(3)/2 - Pi^2/12 + Sum_{n>=1} 1/(n^2*3^n), where Sum_{n>=1} 1/(n^2*3^n) = 0.3662132299770634876167462976642627638...

A307907 a(n) is the greatest k such that p^k <= n for any prime factor p of n.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 3, 1, 1, 2, 1, 5, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 3, 2, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 1, 1, 1, 1, 2, 4, 1, 1, 2, 1, 1, 1, 1

Offset: 2

Rémy Sigrist, May 05 2019

nonn

			For n = 12:
- the prime factors of 12 are 2 and 3,
- 2^2 < 3^2 <= 12 < 3^3,
- hence a(12) = 2.

Rémy Sigrist, Colored scatterplot of the ordinal transform of n -> a(n^5) - 5*a(n) for n = 2..100000

Cf. A006530, A048098, A064052, A090081, A307908.

Array[If[PrimeQ@ #, 1, Floor@ Log[FactorInteger[#][[-1, 1]], #]] &, 105, 2] (* Michael De Vlieger, May 08 2019 *)

a(n) = my (f=factor(n)); logint(n, f[#f~, 1])

from sympy import integer_log, primefactors
def A307907(n): return integer_log(n,max(primefactors(n)))[0] # Chai Wah Wu, Oct 12 2024

a(n) = floor(log(n)/log(A006530(n))).
a(p^k) = k for any prime number p and any k > 0.
0 <= a(n^k) - k*a(n) < k for any n > 1 and any k > 0.
a(n) = 1 iff n belongs to A064052.
a(n) > 1 iff n belongs to A048098.
a(n) > 2 iff n belongs to A090081.

cp's OEIS Frontend

A175475 Decimal expansion of the Dickman function evaluated at 1/3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula