A179671 Products of the 5th power of a prime and a distinct prime of the 3rd power (p^5*q^3).

864, 1944, 4000, 10976, 25000, 30375, 42592, 70304, 83349, 84375, 134456, 157216, 219488, 323433, 389344, 453789, 533871, 780448, 953312, 1071875, 1193859, 1288408, 1620896, 1666737, 2100875, 2205472, 2544224, 2956581, 2970344, 3322336, 4159375, 4348377

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Jul 23 2010

Keywords

• nonn

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Index to sequences related to prime signature

Crossrefs

Cf. A085541, A085965, A085968.

Programs

Mathematica

f[n_]:=Sort[Last/@FactorInteger[n]]=={3,5}; Select[Range[10^6], f]

Python

from sympy import primepi, integer_nthroot, primerange
def A179671(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(integer_nthroot(x//p**5,3)[0]) for p in primerange(integer_nthroot(x,5)[0]+1))+primepi(integer_nthroot(x,8)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

Formula

Sum_{n>=1} 1/a(n) = P(3)*P(5) - P(8) = A085541 * A085965 - A085968 = 0.002187..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020