A295084 - OEIS

%I #28 Oct 06 2024 09:16:20

%S 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,

%T 17,18,18,19,19,19,19,20,20,20,20,21,21,21,21,21,22,22,22,23,30,31,31,

%U 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

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

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

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Smooth_number">Smooth number</a>

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

%F 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).

%F a(n) = (1 - log(2))*n + O(n/log(n)) as n -> infinity. - _Robert Israel_, Nov 14 2017

%p N:= 100: # to get a(1)..a(N)

%p G:= [0,seq(max(numtheory:-factorset(n)),n=2..N)]:

%p seq(nops(select(t -> t^2 <= n, G[1..n])),n=1..N); # _Robert Israel_, Nov 14 2017

%p a:=[];

%p for n from 1 to 100 do

%p    c:=0;

%p    for m from 1 to n do

%p       if A006530(m)^2 <= n then c:=c+1; fi; od:

%p a:=[op(a),c];

%p od:

%p a; # (Included because variants of it will apply to related sequences) - _N. J. A. Sloane_, Apr 10 2020

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

%o (Python)

%o from math import isqrt

%o from sympy import primerange

%o def A295084(n): return int(n-sum(n//p for p in primerange(isqrt(n)+1,n+1))) # _Chai Wah Wu_, Oct 06 2024

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

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

%K nonn,look,changed

%O 1,4

%A _Max Alekseyev_, Nov 13 2017