A179672 Products of the 6th power of a prime and 2 distinct primes (p^6*q*r).

960, 1344, 2112, 2240, 2496, 3264, 3520, 3648, 4160, 4416, 4928, 5440, 5568, 5824, 5952, 6080, 7104, 7290, 7360, 7616, 7872, 8256, 8512, 9024, 9152, 9280, 9920, 10176, 10206, 10304, 11328, 11712, 11840, 11968, 12864, 12992, 13120, 13376, 13632

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

Programs

Mathematica

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,6}; Select[Range[20000], f]

PARI

list(lim)=my(v=List(),t1,t2);forprime(p=2, (lim\6)^(1/6), t1=p^6;forprime(q=2, lim\t1, if(p==q, next);t2=t1*q;forprime(r=q+1, lim\t2, if(p==r,next);listput(v,t2*r)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011

Python

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A179672(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:=x//r**6)))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1)) for r 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])
    return bisection(f,n,n) # Chai Wah Wu, Mar 27 2025