A118092 Odd primes raised to odd prime powers.

27, 125, 243, 343, 1331, 2187, 2197, 3125, 4913, 6859, 12167, 16807, 24389, 29791, 50653, 68921, 78125, 79507, 103823, 148877, 161051, 177147, 205379, 226981, 300763, 357911, 371293, 389017, 493039, 571787, 704969, 823543, 912673, 1030301

Offset: 1

Author

Jonathan Vos Post, May 11 2006

Keywords

• easy
• nonn

Comments

Subset of A053810 Prime powers of prime numbers. Subset of A000961 Prime powers. Subsets include A030078 Cubes of primes, A050997 Fifth powers of primes.

Crossrefs

Cf. A000040, A000961, A030078, A050997, A051006, A053810, A065091.

Programs

Mathematica

With[{prs=Prime[Range[2,30]]},Take[Union[First[#]^Last[#]&/@ Tuples[prs,2]],40]] (* Harvey P. Dale, Dec 23 2011 *)

Python

from sympy import primepi, integer_nthroot, primerange
def A118092(n):
    def f(x): return int(n+x-sum(primepi(integer_nthroot(x, p)[0])-1 for p in primerange(3,x.bit_length())))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f,n,n) # Chai Wah Wu, Sep 12 2024

Formula

{p^q where p is in A065091 and q is in A065091}.

Sum_{n>=1} 1/a(n) = Sum_{p odd prime} P(p) - A051006 + 1/4 = 0.054745292329555814476..., where P(s) is the prime zeta function. - Amiram Eldar, Sep 13 2024

Extensions

Extended by Ray Chandler, Oct 28 2008