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A267697 Numbers with 7 odd divisors.

%I A267697 #40 Feb 22 2025 01:23:07
%S A267697 729,1458,2916,5832,11664,15625,23328,31250,46656,62500,93312,117649,
%T A267697 125000,186624,235298,250000,373248,470596,500000,746496,941192,
%U A267697 1000000,1492992,1771561,1882384,2000000,2985984,3543122,3764768,4000000,4826809,5971968,7086244,7529536,8000000,9653618
%N A267697 Numbers with 7 odd divisors.
%C A267697 Positive integers that have exactly seven odd divisors.
%C A267697 Numbers k such that the symmetric representation of sigma(k) has 7 subparts. - _Omar E. Pol_, Dec 28 2016
%C A267697 Numbers that can be formed in exactly 6 ways by summing sequences of 2 or more consecutive positive integers. - _Julie Jones_, Aug 13 2018
%C A267697 Numbers of the form p^6 * 2^k where p is an odd prime. - _David A. Corneth_, Aug 14 2018
%H A267697 David A. Corneth, <a href="/A267697/b267697.txt">Table of n, a(n) for n = 1..10000</a>
%F A267697 A001227(a(n)) = 7.
%F A267697 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
%o A267697 (PARI) isok(n) = sumdiv(n, d, (d%2)) == 7; \\ _Michel Marcus_, Apr 03 2016
%o A267697 (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
%o A267697 (Python)
%o A267697 from sympy import integer_log, primerange, integer_nthroot
%o A267697 def A267697(n):
%o A267697     def bisection(f,kmin=0,kmax=1):
%o A267697         while f(kmax) > kmax: kmax <<= 1
%o A267697         kmin = kmax >> 1
%o A267697         while kmax-kmin > 1:
%o A267697             kmid = kmax+kmin>>1
%o A267697             if f(kmid) <= kmid:
%o A267697                 kmax = kmid
%o A267697             else:
%o A267697                 kmin = kmid
%o A267697         return kmax
%o A267697     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)))
%o A267697     return bisection(f,n,n) # _Chai Wah Wu_, Feb 22 2025
%Y A267697 Column 7 of A266531.
%Y A267697 Cf. A001227, A030516, A038547, A085966, A236104, A237593, A279387.
%Y A267697 Numbers with k odd divisors (k = 1..10): A000079, A038550, A072502, apparently A131651, A267696, A230577, this sequence, A267891, A267892, A267893.
%K A267697 nonn,easy
%O A267697 1,1
%A A267697 _Omar E. Pol_, Apr 03 2016
%E A267697 More terms from _Michel Marcus_, Apr 03 2016