A242493 a(n) is the number of not-sqrt-smooth numbers ("jagged" numbers) not exceeding n. This is the counting function of A064052.
0, 1, 2, 2, 3, 4, 5, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 11, 12, 13, 14, 15, 16, 16, 16, 17, 17, 18, 19, 19, 20, 20, 21, 22, 23, 23, 24, 25, 26, 26, 27, 28, 29, 30, 30, 31, 32, 32, 32, 32, 33, 34, 35, 35, 36, 36, 37, 38, 39, 39, 40, 41, 41, 41, 42, 43, 44, 45
Offset: 1
Keywords
References
- Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, chapter 2.21, p. 166.
- Daniel H. Greene and Donald E. Knuth, Mathematics for the Analysis of Algorithms, 3rd ed., Birkhäuser, 1990, pp. 95-98.
Programs
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Mathematica
jaggedQ[n_] := jaggedQ[n] = (f = FactorInteger[n][[All, 1]]; s = Sqrt[n]; Count[f, p_ /; p > s] > 0); a[n_] := ( For[ cnt = 0; j = 2, j <= n, j++, If[jaggedQ[j], cnt++]]; cnt); Table[a[n], {n, 1, 100}] -
Python
from math import isqrt from sympy import primepi def A242493(n): return sum(primepi(n//i)-primepi(i) for i in range(1,isqrt(n)+1)) # Chai Wah Wu, Sep 01 2024
Formula
From Ridouane Oudra, Nov 07 2019: (Start)
a(n) = Sum_{i=1..floor(sqrt(n))} (pi(floor(n/i)) - pi(i)).
a(n) = Sum_{p<=sqrt(n)} (p-1) + Sum_{sqrt(n)
a(n) = n - A064775(n). (End)
a(n) ~ log(2)*n - A153810 * n/log(n) - A242610 * n/log(n)^2 + O(n/log(n)^3) (Greene and Knuth, 1990). - Amiram Eldar, Apr 15 2021
A386718 Decimal expansion of Integral_{x=0..1} Integral_{y=0..1} Integral_{z=0..1} {1/(x*y*z)} dx dy dz, where {} denotes fractional part.
5, 0, 0, 4, 4, 5, 3, 6, 2, 1, 7, 8, 5, 8, 0, 0, 2, 3, 4, 9, 6, 3, 3, 9, 4, 7, 8, 8, 1, 0, 1, 0, 5, 1, 5, 2, 7, 7, 5, 1, 0, 9, 9, 0, 5, 4, 4, 5, 0, 8, 4, 7, 2, 8, 7, 3, 3, 5, 9, 0, 0, 0, 7, 5, 8, 2, 4, 5, 9, 0, 8, 4, 4, 8, 4, 9, 8, 7, 0, 2, 1, 0, 2, 7, 1, 2, 8, 9, 6, 3, 6, 4, 3, 7, 8, 4, 5, 3, 3, 7, 4, 9, 0, 8, 8
Offset: 0
Examples
0.50044536217858002349633947881010515277510990544508...
References
- Ovidiu Furdui, Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis, New York: Springer, 2013. See section 2.43, page 106.
Links
- Yaming Yu, A Multiple Integral in Terms of Stieltjes Constants, SIAM Problems and Solutions, Classical Analysis, Integrals, Problem 07-002 (2007).
Crossrefs
Programs
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Mathematica
With[{m = 2}, RealDigits[1 - Sum[StieltjesGamma[k]/k!, {k, 0, 2}], 10, 120][[1]]]
Formula
Equals 1 - gamma - gamma_1 - gamma_2/2, where gamma_k is the k-th Stieltjes constant.
In general, for m >= 1, Integral_{x_1=0..1} ... Integral_{x_m=0..1} {1/(x_1*...*x_m)} dx_1 ... dx_m = 1 - Sum_{k=0..m-1} gamma_k/k!, where gamma_0 = gamma is Euler's constant.
Comments