A179667 Products of the 5th power of a prime and 2 distinct primes (p^5*q*r).

480, 672, 1056, 1120, 1248, 1632, 1760, 1824, 2080, 2208, 2430, 2464, 2720, 2784, 2912, 2976, 3040, 3402, 3552, 3680, 3808, 3936, 4128, 4256, 4512, 4576, 4640, 4960, 5088, 5152, 5346, 5664, 5856, 5920, 5984, 6318, 6432, 6496, 6560, 6688, 6816, 6880

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, Prime Signatures
• Index to sequences related to prime signature

Programs

Mathematica

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,5}; Select[Range[10000], f]

PARI

list(lim)=my(v=List(),t);forprime(p=2,(lim\6)^(1/5),forprime(q=2,sqrt(lim\p^5),if(p==q,next);t=p^5*q;forprime(r=q+1,lim\t,if(p==r,next);listput(v,t*r))));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011

Python

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A179667(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**5)))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1)) for r in primerange(integer_nthroot(x,5)[0]+1))+sum(primepi(x//p**6) for p in primerange(integer_nthroot(x,6)[0]+1))-primepi(integer_nthroot(x,7)[0])
    return bisection(f,n,n) # Chai Wah Wu, Mar 27 2025