A179691 Numbers p^5*q^2*r where p, q, r are 3 distinct primes.

1440, 2016, 2400, 3168, 3744, 4704, 4860, 4896, 5472, 5600, 6624, 6804, 7840, 8352, 8800, 8928, 10400, 10656, 10692, 11616, 11808, 12150, 12384, 12636, 13536, 13600, 15200, 15264, 16224, 16524, 16992, 17248, 17568, 18400, 18468, 19296, 19360

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Jul 24 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

Part of the list A178739 .. A179696 (and A030514 .. A030629, A189982 .. A189990 etc, cf. A101296). - M. F. Hasler, Jul 17 2019

Subsequence of A175746 (numbers with 36 divisors).

Programs

Mathematica

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,2,5}; Select[Range[20000], f]

PARI

list(lim)=my(v=List(),t1,t2);forprime(p=2, (lim\12)^(1/5), t1=p^5;forprime(q=2, sqrt(lim\t1), if(p==q, next);t2=t1*q^2;forprime(r=2, lim\t2, if(p==r||q==r, next);listput(v,t2*r)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 24 2011

Python

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A179691(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(x//(p**5*q**2)) for p in primerange(integer_nthroot(x,5)[0]+1) for q in primerange(isqrt(x//p**5)+1))+sum(primepi(integer_nthroot(x//p**5,3)[0]) for p in primerange(integer_nthroot(x,5)[0]+1))+sum(primepi(isqrt(x//p**6)) for p in primerange(integer_nthroot(x,6)[0]+1))+sum(primepi(x//p**7) for p in primerange(integer_nthroot(x,7)[0]+1))-(primepi(integer_nthroot(x,8)[0])<<1)
    return bisection(f,n,n) # Chai Wah Wu, Mar 27 2025

Extensions

Name improved by M. F. Hasler, Jul 17 2019