A085501 Number of prime powers p^k <= n that are not prime (k = 0 or k > 1).

1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11

Offset: 1

Author

Reinhard Zumkeller, Jul 03 2003

Keywords

• nonn
• easy

Comments

a(n) = Max{k: A025475(k)<=n};

a(n)=A065515(n)-A000720(n)=A069637(n)+1;

for n<36=(2*3)^2: a(n) = A069623(n).

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Prime Power.
• Eric Weisstein's World of Mathematics, Perfect Powers.

Programs

PARI

a(n)=sum(k=2,logint(n,2), primepi(sqrtnint(n,k)))+1 \\ Charles R Greathouse IV, Jul 21 2017

PARI

first(n)=my(v=vector(n),s=1); for(e=2,logint(n,2), forprime(p=2,sqrtnint(n,e), v[p^e]=1)); for(i=1,n, s+=v[i]; v[i]=s); v \\ Charles R Greathouse IV, Jul 21 2017

Python

from sympy import primepi, integer_nthroot
def A085501(n): return 1+sum(primepi(integer_nthroot(n,k)[0]) for k in range(2,n.bit_length())) # Chai Wah Wu, Aug 15 2024