A267697 Numbers with 7 odd divisors.

729, 1458, 2916, 5832, 11664, 15625, 23328, 31250, 46656, 62500, 93312, 117649, 125000, 186624, 235298, 250000, 373248, 470596, 500000, 746496, 941192, 1000000, 1492992, 1771561, 1882384, 2000000, 2985984, 3543122, 3764768, 4000000, 4826809, 5971968, 7086244, 7529536, 8000000, 9653618

Offset: 1

Author

Omar E. Pol, Apr 03 2016

Keywords

• nonn
• easy

Comments

Positive integers that have exactly seven odd divisors.

Numbers k such that the symmetric representation of sigma(k) has 7 subparts. - Omar E. Pol, Dec 28 2016

Numbers that can be formed in exactly 6 ways by summing sequences of 2 or more consecutive positive integers. - Julie Jones, Aug 13 2018

Numbers of the form p^6 * 2^k where p is an odd prime. - David A. Corneth, Aug 14 2018

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000

Crossrefs

Column 7 of A266531.

Cf. A001227, A030516, A038547, A085966, A236104, A237593, A279387.

Numbers with k odd divisors (k = 1..10): A000079, A038550, A072502, apparently A131651, A267696, A230577, this sequence, A267891, A267892, A267893.

Programs

PARI

isok(n) = sumdiv(n, d, (d%2)) == 7; \\ Michel Marcus, Apr 03 2016

PARI

upto(n) = {my(res = List()); forprime(p = 3, sqrtnint(n, 6), listput(res, p^6)); q = #res; for(i = 1, q, odd = res[i]; for(j = 1, logint(n \ odd, 2), listput(res, odd <<= 1))); listsort(res); res} \\ David A. Corneth, Aug 14 2018

Python

from sympy import integer_log, primerange, integer_nthroot
def A267697(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 int(n+x-sum(integer_log(x//p**6,2)[0]+1 for p in primerange(3,integer_nthroot(x,6)[0]+1)))
    return bisection(f,n,n) # Chai Wah Wu, Feb 22 2025

Formula

A001227(a(n)) = 7.

Sum_{n>=1} 1/a(n) = 2 * P(6) - 1/32 = 0.00289017370127..., where P(6) is the value of the prime zeta function at 6 (A085966). - Amiram Eldar, Sep 16 2024

Extensions

More terms from Michel Marcus, Apr 03 2016