A295101 Number of squarefree sqrt(n)-smooth numbers <= n.

1, 1, 1, 2, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14

Offset: 1

Author

Max Alekseyev, Nov 14 2017

Keywords

• nonn
• look

Comments

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

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A005117, A013928, A295084.

Programs

Maple

N:= 200: # for a(1)..a(N)
V:= Vector(N,1):
for n from 2 to N do
   if not numtheory:-issqrfree(n) then next fi;
   m:= max(max(numtheory:-factorset(n))^2,n);
   if m <= N then V[m..N]:= map(`+`,V[m..N],1) fi;
od:
convert(V,list); # Robert Israel, Mar 24 2020

Formula

a(n) = A013928(n+1) - Sum_{prime p > sqrt(n)} A013928(floor(n/p)+1).

If n is in A295102, 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)+A013928(p); otherwise a(n)=a(n-1).