A267574 Number of nontrivial prime powers p^k (k>1) less than 10^n.

3, 10, 25, 51, 108, 236, 555, 1404, 3689, 10084, 28156, 80070, 230567, 670121, 1962689, 5782467, 17124205, 50930439, 152043591, 455389239, 1367883343, 4119448336, 12434731101, 37613760489, 113995567274, 346090346046, 1052421430208, 3205047877403, 9774085385959, 29845027519170, 91239740502962, 279240320955782, 855506687516860, 2623565774949376

Offset: 1

Author

Daniel Mondot, Jan 17 2016

Keywords

• nonn

Comments

Computed up to 10^19 by program. The program was written in C, and is rather long. It starts by finding all prime numbers up to 4*10^9, then uses that to count all powers of these primes up to 10^19.

Examples

			For n=1, there are 3 powers of prime numbers less than 10^1: 2^2, 2^3 and 3^2. i.e. 4, 8, 9.
For n=2, there are 10 powers of prime numbers less than 10^2: 4, 8, 9, 16, 25, 27, 32, 49, 64, 81.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..48
• Daniel Mondot, Math puzzle about Least Common Multiple (french:PPCM)

Crossrefs

Cf. A246547, A006880.

Programs

Mathematica

a[n_] := Sum[PrimePi [10^(n/k)], {k, 2, n * Log2[10]}]; Array[a, 20] (* Giovanni Resta, Apr 09 2016 *)

Python

from sympy import primepi, integer_nthroot
def A267574(n): return int(sum(primepi(integer_nthroot(10**n,k)[0]) for k in range(2,(10**n).bit_length()))) # Chai Wah Wu, Aug 14 2024

SageMath

def A267574(n):
    gen = (p for p in srange(2, 10^n) if p.is_prime_power() and not p.is_prime())
    return sum(1 for _ in gen)
print([A267574(n) for n in range(1, 7)])  # Peter Luschny, Sep 16 2023

Extensions

a(20)-a(26) from Chai Wah Wu, Jan 25 2016

a(27)-a(34) from Giovanni Resta, Apr 09 2016