A179666 Products of the 4th power of a prime and a distinct prime of power 3 (p^4*q^3).

432, 648, 2000, 5000, 5488, 10125, 16875, 19208, 21296, 27783, 35152, 64827, 78608, 107811, 109744, 117128, 177957, 194672, 214375, 228488, 300125, 390224, 395307, 397953, 476656, 555579, 668168, 771147, 810448, 831875

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
• Will Nicholes, List of Prime Signatures
• Index to sequences related to prime signature

Crossrefs

Cf. A046308, A030638, A007774.

Cf. A085541, A085964, A085967.

Programs

Mathematica

f[n_]:=Sort[Last/@FactorInteger[n]]=={3,4}; Select[Range[10^6], f]
With[{nn=40},Select[Flatten[{#[[1]]^4 #[[2]]^3,#[[1]]^3 #[[2]]^4}&/@ Subsets[ Prime[Range[nn]],{2}]]//Union,#<=16nn^3&]] (* Harvey P. Dale, Nov 15 2020 *)

PARI

list(lim)=my(v=List(),t);forprime(p=2,(lim\8)^(1/4),t=p^4;forprime(q=2,(lim\t)^(1/3),if(p==q,next);listput(v,t*q^3)));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011

Python

from sympy import primepi, integer_nthroot, primerange
def A179666(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**4,3)[0]) for p in primerange(integer_nthroot(x,4)[0]+1))+primepi(integer_nthroot(x,7)[0])
    return bisection(f,n,n) # Chai Wah Wu, Mar 27 2025

Formula

Sum_{n>=1} 1/a(n) = P(3)*P(4) - P(7) = A085541 * A085964 - A085967 = 0.005171..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020