A179695 Numbers of the form p^3*q^2*r^2 where p, q, and r are distinct primes.

1800, 2700, 3528, 4500, 5292, 8712, 9800, 12168, 12348, 13068, 18252, 20808, 24200, 24500, 25992, 31212, 33075, 33800, 34300, 38088, 38988, 47432, 47916, 55125, 57132, 57800, 60500, 60552, 66248, 69192, 72200, 77175, 79092, 81675, 84500

Offset

1,1

Comments

Subsequence of A225228. - Reinhard Zumkeller, May 03 2013

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

Will Nicholes, List of prime signatures, 2010.

Index to sequences related to prime signature.

Formula

A050326(a(n)) = 5. - Reinhard Zumkeller, May 03 2013

Sum_{n>=1} 1/a(n) = P(2)^2*P(3)/2 - P(3)*P(4)/2 - P(2)*P(5) + P(7) = 0.0032578591481263202818..., where P is the prime zeta function. - Amiram Eldar, Mar 07 2024

Mathematica

f[n_]:=Sort[Last/@FactorInteger[n]]=={2, 2, 3}; Select[Range[10^5], f]
f[n_]:={Times@@(n^{2, 2, 3}), Times@@(n^{2, 3, 2}), Times@@(n^{3, 2, 2})}; Module[ {nn=20}, Select[Flatten[f/@Subsets[Prime[Range[nn]], {3}]], #<= 72*Prime[ nn]^2&]]//Union (* Harvey P. Dale, Jul 05 2019 *)

Prog

(PARI) list(lim)=my(v=List(), t1, t2); forprime(p=2, (lim\36)^(1/3), t1=p^3; forprime(q=2, sqrt(lim\t1), if(p==q, next); t2=t1*q^2; forprime(r=q+1, sqrt(lim\t2), if(p==r, next); listput(v, t2*r^2)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 24 2011
(Python)
from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A179695(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((t:=primepi(s:=isqrt(y:=isqrt(x//r**3))))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1)) for r in primerange(integer_nthroot(x, 3)[0]+1))+sum(primepi(isqrt(x//p**5)) for p in primerange(integer_nthroot(x, 5)[0]+1))-primepi(integer_nthroot(x, 7)[0])
    return bisection(f, n, n) # Chai Wah Wu, Mar 27 2025

Crossrefs

Cf. A050326, A225228.

Cf. A085548, A085541, A085964, A085965, A085967.

Sequence in context: A394000 A394050 A392927 * A375013 A380847 A025139

Adjacent sequences: A179692 A179693 A179694 * A179696 A179697 A179698

Keyword

nonn

Author

Vladimir Joseph Stephan Orlovsky, Jul 24 2010

Status

approved