A280609 Odd prime powers with prime exponents.

9, 25, 27, 49, 121, 125, 169, 243, 289, 343, 361, 529, 841, 961, 1331, 1369, 1681, 1849, 2187, 2197, 2209, 2809, 3125, 3481, 3721, 4489, 4913, 5041, 5329, 6241, 6859, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12167, 12769, 16129, 16807, 17161, 18769, 19321, 22201, 22801, 24389, 24649, 26569, 27889, 29791, 29929

Offset

1,1

Comments

Intersection of A053810 and A061345.

Links

Table of n, a(n) for n=1..53.

Eric Weisstein's World of Mathematics, Prime Power.

Formula

a(n) = p^q, where p, q are primes and p > 2.

Sum_{n>=1} 1/a(n) = Sum_{p prime} P(p) - A051006 = 0.25699271237062131298..., where P(s) is the prime zeta function. - Amiram Eldar, Sep 13 2024

Example

9 is in the sequence because 9 = 3^2;
25 is in the sequence because 25 = 5^2;
27 is in the sequence because 27 = 3^3, etc.

Mathematica

Select[Range[30000], PrimePowerQ[#1] && PrimeQ[PrimeOmega[#1]] && Mod[#1, 2] == 1 & ]

Prog

(Python)
from sympy import primepi, integer_nthroot, primerange
def A280609(n):
    def f(x): return int(n+x-sum(primepi(integer_nthroot(x, p)[0])-1 for p in primerange(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

Crossrefs

Cf. A000961, A034785, A051006, A053810, A061345, A078422, A246547, A246551, A246655.

Sequence in context: A244623 A339127 A117580 * A340238 A020308 A108989

Adjacent sequences: A280606 A280607 A280608 * A280610 A280611 A280612

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Jan 06 2017

Status

approved