A242493 -id:A242493 - cp's OEIS frontend

A064775 Number of positive integers k <= n such that all prime divisors of k are <= sqrt(k).

1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 7, 8, 9, 9, 10, 10, 10, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 16, 17, 18, 18, 18, 18, 19, 19, 20, 20, 20, 20, 21, 21, 21, 22, 23, 23, 23, 23, 23, 23, 24, 24, 25, 25, 25, 26, 26, 26, 26

Offset: 1

Benoit Cloitre, May 11 2002

A048098(n) is the n-th number k such that all prime divisors of k are <= sqrt(k).

			Below 28, only k=27,25,24,18,16,12,9,8,4,1 have all their prime divisors less than or equal to sqrt(k), hence a(28)=10. To obtain from A048098(n): A048098(10) = 27 <= 28 < A048098(11)=30, hence a(28)=10.

D. P. Parent, Exercices de théorie des nombres, Les grands classiques, Gauthier-Villars, Edition Jacques Gabay, p. 17.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A048098, A242493.

The following are all different versions of sqrt(n)-smooth numbers: A048098, A063539, A064775, A295084, A333535, A333536.

[1] cat [#[k:k in [1..n]|forall{p:p in PrimeDivisors(k)| p le Sqrt(k)}]: n in [2..80]]; // Marius A. Burtea, Nov 08 2019

a(n)=n-sum(k=1,sqrtint(n),(k-1)*isprime(k)) - sum(k=sqrtint(n)+1, n, floor(n/k)*isprime(k))

from math import isqrt
from sympy import primepi
def A064775(n): return int(n+sum(primepi(i)-primepi(n//i) for i in range(1,isqrt(n)+1))) # Chai Wah Wu, Oct 05 2024

a(n) = n - (Sum_{p<=sqrt(n)} (p-1)) - Sum_{sqrt(n)A048098(k) <= n. Asymptotically: a(n) = (1-log(2))*n + O(n/log(n)).
From Ridouane Oudra, Nov 07 2019: (Start)
a(n) = n - Sum_{i=1..floor(sqrt(n))} (pi(floor(n/i)) - pi(i)).
a(n) = n - A242493(n). (End)

A242610 Decimal expansion of 1-gamma-gamma(1), a constant related to the asymptotic expansion of j(n), the counting function of "jagged" numbers, where gamma is Euler-Mascheroni constant and gamma(1) the first Stieltjes constant.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, May 19 2014

			0.495600180582143864254074285792498888...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, chapter 2.21, p. 166.
Ovidiu Furdui, Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis, New York: Springer, 2013. See Problem 2.17, p. 102.

Ovidiu Furdui, Problem 164, Missouri J. Math. Sci., Vol. 18, No. 2 (2006), p. 148; Solution, ibid., Vol. 19, No. 2 (2007), pp. 156-158.

Cf. A064052, A082633, A153810, A242493.

RealDigits[1 - EulerGamma - StieltjesGamma[1], 10, 100] // First

j(n) = log(2)*n - (1-gamma)*n/log(n) - (1-gamma-gamma(1))*n/log(n)^2 + O(n/log(n)^3).
Equals -Integral_{x=0..1} frac(1/x)*log(x) dx (Furdui, 2007 and 2013). - Amiram Eldar, Mar 26 2022
Equals Integral_{x=0..1} Integral_{y=0..1} frac(1/(x*y)) dx dy (Furdui, 2013, section 2.43, p. 106). - Amiram Eldar, Jul 31 2025

cp's OEIS Frontend

A064775 Number of positive integers k <= n such that all prime divisors of k are <= sqrt(k).

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