cp's OEIS Frontend

A030627 Numbers with 9 divisors.

%I A030627 #29 Feb 21 2025 16:47:01
%S A030627 36,100,196,225,256,441,484,676,1089,1156,1225,1444,1521,2116,2601,
%T A030627 3025,3249,3364,3844,4225,4761,5476,5929,6561,6724,7225,7396,7569,
%U A030627 8281,8649,8836,9025,11236,12321,13225,13924,14161,14884,15129
%N A030627 Numbers with 9 divisors.
%C A030627 Numbers of the form p^8 (8th row of A120458) or p^2*r^2 (A085986), where p and r are distinct primes. - _R. J. Mathar_, Mar 01 2010
%H A030627 R. J. Mathar, <a href="/A030627/b030627.txt">Table of n, a(n) for n = 1..1000</a>
%F A030627 A000005(a(n)) = 9. - _Juri-Stepan Gerasimov_, Oct 10 2009
%F A030627 Sum_{n>=1} 1/a(n) = (P(2)^2 - P(4))/2 + P(8) = 0.0678286..., where P is the prime zeta function. - _Amiram Eldar_, Jul 03 2022
%t A030627 Select[Range[90000],DivisorSigma[0,#]==9&] (* _Vladimir Joseph Stephan Orlovsky_, May 05 2011 *)
%o A030627 (PARI) is(n)=numdiv(n)==9 \\ _Charles R Greathouse IV_, Jun 19 2016
%o A030627 (Python)
%o A030627 from math import isqrt
%o A030627 from sympy import primepi, primerange
%o A030627 def A030627(n):
%o A030627     def bisection(f,kmin=0,kmax=1):
%o A030627         while f(kmax) > kmax: kmax <<= 1
%o A030627         kmin = kmax >> 1
%o A030627         while kmax-kmin > 1:
%o A030627             kmid = kmax+kmin>>1
%o A030627             if f(kmid) <= kmid:
%o A030627                 kmax = kmid
%o A030627             else:
%o A030627                 kmin = kmid
%o A030627         return kmax
%o A030627     def f(x): return int(n+x+(t:=primepi(s:=isqrt(y:=isqrt(x))))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1))-primepi(isqrt(s)))
%o A030627     return bisection(f,n,n) # _Chai Wah Wu_, Feb 21 2025
%Y A030627 Cf. A000005, A030515, A030516, A030626, A085986, A120458, A179645.
%K A030627 nonn
%O A030627 1,1
%A A030627 _Jeff Burch_