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

Author

Jean-François Alcover, May 16 2014

Keywords

• nonn

Comments

This sequence is different from shifted A072490, after 22 terms.

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.

Crossrefs

Cf. A064052, A064775, A072490, A153810, A242610.

Programs

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

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.