A267712 Number of nontrivial prime powers p^k (k > 0) less than 10^n.

7, 35, 193, 1280, 9700, 78734, 665134, 5762859, 50851223, 455062595, 4118082969, 37607992088, 346065767406, 3204942420923, 29844572385358, 279238346816392, 2623557174778438, 24739954338671299, 234057667428388198, 2220819603016308079, 21127269487386615271, 201467286693435354626, 1925320391619238700024, 18435599767386814628355, 176846309399257764978954, 1699246750872783231673649

Offset: 1

Author

Daniel Mondot, Jan 19 2016

Keywords

• nonn

Comments

This is the sum of A006880 and A267574, term by term.

Examples

			For n=1, there are 4 primes plus 3 prime powers less than 10^1: 2, 3, 4, 5, 7, 8, 9; 7 in total.
		

Crossrefs

Cf. A006880, A238815, A267574, A089579.

Programs

Mathematica

Table[Count[Range[10^n], k_ /; PrimePowerQ@ k], {n, 6}] (* Michael De Vlieger, Jan 20 2016 *)
f[n_] := Sum[PrimePi[10^(n/k)], {k, n*Log2[10]}]; Array[f, 14] (* Robert G. Wilson v, Aug 17 2017, after Giovanni Resta in A267574 *)

SageMath

def A267712(n):
    gen = (p for p in srange(2, 10^n) if p.is_prime_power())
    return sum(1 for _ in gen)
print([A267712(n) for n in range(1, 7)])  # Peter Luschny, Sep 16 2023

Formula

a(n) = A006880(n) + A267574(n).

a(n) = A238815(n) - 1. - Jon E. Schoenfield, Apr 19 2018

Extensions

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