A295084 Number of sqrt(n)-smooth numbers <= n.

1, 1, 1, 3, 3, 3, 3, 4, 7, 7, 7, 8, 8, 8, 8, 9, 9, 10, 10, 10, 10, 10, 10, 11, 16, 16, 17, 17, 17, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 21, 22, 22, 22, 23, 30, 31, 31, 31, 31, 32, 32, 33, 33, 33, 33, 34, 34, 34, 35, 36, 36, 36, 36, 36, 36, 37, 37, 38, 38, 38, 39, 39, 39, 39, 39, 40

Offset: 1

Author

Max Alekseyev, Nov 13 2017

Keywords

• nonn
• look

Comments

a(n) = number of positive integers m<=n such that A006530(m) <= sqrt(n).

Links

• Wikipedia, Smooth number

Crossrefs

Cf. A048098 (indices of records), A063539, A241419.

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

Programs

Maple

N:= 100: # to get a(1)..a(N)
G:= [0,seq(max(numtheory:-factorset(n)),n=2..N)]:
seq(nops(select(t -> t^2 <= n, G[1..n])),n=1..N); # Robert Israel, Nov 14 2017
a:=[];
for n from 1 to 100 do
   c:=0;
   for m from 1 to n do
      if A006530(m)^2 <= n then c:=c+1; fi; od:
a:=[op(a),c];
od:
a; # (Included because variants of it will apply to related sequences) - N. J. A. Sloane, Apr 10 2020

PARI

A295084(n) = my(r=n); forprime(p=sqrtint(n)+1,n, r-=n\p); r;

Python

from math import isqrt
from sympy import primerange
def A295084(n): return int(n-sum(n//p for p in primerange(isqrt(n)+1,n+1))) # Chai Wah Wu, Oct 06 2024

Formula

a(n) = n - A241419(n).

If n is in A063539, then a(n)=a(n-1)+1; if n is in A001248, i.e., n=p^2 for prime p, then a(n)=a(n-1)+p; otherwise a(n)=a(n-1).

a(n) = (1 - log(2))*n + O(n/log(n)) as n -> infinity. - Robert Israel, Nov 14 2017