A030632 Numbers with 14 divisors.

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, 8192, 8384, 8768, 8896, 9477, 9536, 9664, 10048, 10432

Offset: 1

Author

Jeff Burch

Keywords

• nonn

Comments

Numbers of the form p^13 (A138031) or p*q^6 (A189987), where p and q are distinct primes. - R. J. Mathar, Mar 01 2010

Links

• R. J. Mathar, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A092759.

Programs

Mathematica

Select[Range[15000], DivisorSigma[0, #] == 14 &]

PARI

is(n)=numdiv(n)==14 \\ Charles R Greathouse IV, Jun 19 2016

Python

from sympy import primepi, primerange, integer_nthroot
def A030632(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])-primepi(integer_nthroot(x,13)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 22 2025