A179646 Product of the 5th power of a prime and different distinct prime of the 2nd power (p^5*q^2).

288, 800, 972, 1568, 3872, 5408, 6075, 9248, 11552, 11907, 12500, 16928, 26912, 28125, 29403, 30752, 41067, 43808, 53792, 59168, 67228, 70227, 70688, 87723, 89888, 111392, 119072, 128547, 143648, 151263, 153125, 161312, 170528, 199712

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Jul 21 2010

Keywords

• nonn

Comments

288=2^5*3^2, 800=2^5*5^2,..

Links

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

Crossrefs

Cf. A030636, A046308, A007774.

Cf. A085548, A085965, A085967.

Programs

Mathematica

f[n_]:=Sort[Last/@FactorInteger[n]]=={2,5}; Select[Range[200000], f]

PARI

list(lim)=my(v=List(),t);forprime(p=2,(lim\4)^(1/5),t=p^5;forprime(q=2,sqrt(lim\t),if(p==q,next);listput(v,t*q^2)));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011

Python

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A189988(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(isqrt(x//p**4)) for p in primerange(integer_nthroot(x,4)[0]+1))+primepi(integer_nthroot(x,6)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

Formula

Sum_{n>=1} 1/a(n) = P(2)*P(5) - P(7) = A085548 * A085965 - A085967 = 0.007886..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020