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

1,1

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

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

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 *)

Prog

(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

Crossrefs

Cf. A046308, A030638, A007774.

Cf. A085541, A085964, A085967.

Sequence in context: A388293 A392908 A231231 * A203663 A250815 A006910

Adjacent sequences: A179663 A179664 A179665 * A179667 A179668 A179669

Keyword

nonn

Author

Vladimir Joseph Stephan Orlovsky, Jul 23 2010

Status

approved