A189987 Numbers with prime factorization p*q^6.

192, 320, 448, 704, 832, 1088, 1216, 1458, 1472, 1856, 1984, 2368, 2624, 2752, 3008, 3392, 3645, 3776, 3904, 4288, 4544, 4672, 5056, 5103, 5312, 5696, 6208, 6464, 6592, 6848, 6976, 7232, 8019, 8128, 8384, 8768, 8896, 9477, 9536, 9664, 10048, 10432, 10688

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, May 03 2011

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

Crossrefs

Cf. A030632, A092759.

Programs

Mathematica

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,6}; Select[Range[30000],f]

PARI

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

Python

from sympy import primepi, primerange, integer_nthroot
def A189987(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**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, Feb 22 2025