cp's OEIS Frontend

A216427 Numbers of the form a^2*b^3, where a >= 2 and b >= 2.

%I A216427 #15 Sep 14 2024 12:31:35
%S A216427 32,72,108,128,200,243,256,288,392,432,500,512,576,648,675,800,864,
%T A216427 968,972,1024,1125,1152,1323,1352,1372,1568,1600,1728,1800,1944,2000,
%U A216427 2048,2187,2304,2312,2592,2700,2888,2916,3087,3125,3136,3200,3267,3456,3528,3872,3888,4000,4096,4232,4500,4563,4608
%N A216427 Numbers of the form a^2*b^3, where a >= 2 and b >= 2.
%C A216427 Powerful numbers (A001694) that are not squares of cubefree numbers (A004709), cubes of squarefree numbers (A062838), or 6th powers of primes (A030516). - _Amiram Eldar_, Feb 07 2023
%H A216427 Charles R Greathouse IV, <a href="/A216427/b216427.txt">Table of n, a(n) for n = 1..10000</a>
%F A216427 Sum_{n>=1} 1/a(n) = 1 + ((zeta(2)-1)*(zeta(3)-1)-1)/zeta(6) - P(6) = 0.12806919584708298724..., where P(s) is the prime zeta function. - _Amiram Eldar_, Feb 07 2023
%t A216427 With[{max = 5000}, Union[Table[i^2*j^3, {j, 2, max^(1/3)}, {i, 2, Sqrt[max/j^3]}] // Flatten]] (* _Amiram Eldar_, Feb 07 2023 *)
%o A216427 (PARI) list(lim)=my(v=List()); for(b=2, sqrtnint(lim\4, 3), for(a=2, sqrtint(lim\b^3), listput(v, a^2*b^3))); Set(v) \\ _Charles R Greathouse IV_, Jan 03 2014
%o A216427 (Python)
%o A216427 from math import isqrt
%o A216427 from sympy import mobius, integer_nthroot, primepi
%o A216427 def A216427(n):
%o A216427     def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
%o A216427     def bisection(f,kmin=0,kmax=1):
%o A216427         while f(kmax) > kmax: kmax <<= 1
%o A216427         while kmax-kmin > 1:
%o A216427             kmid = kmax+kmin>>1
%o A216427             if f(kmid) <= kmid:
%o A216427                 kmax = kmid
%o A216427             else:
%o A216427                 kmin = kmid
%o A216427         return kmax
%o A216427     def f(x):
%o A216427         j, b = isqrt(x), integer_nthroot(x,6)[0]
%o A216427         l, c = 0, n+x-1+primepi(b)+sum(mobius(k)*(j//k**3) for k in range(1, b+1))
%o A216427         while j>1:
%o A216427             k2 = integer_nthroot(x//j**2,3)[0]+1
%o A216427             w = squarefreepi(k2-1)
%o A216427             c -= j*(w-l)
%o A216427             l, j = w, isqrt(x//k2**3)
%o A216427         return c+l
%o A216427     return bisection(f,n,n) # _Chai Wah Wu_, Sep 13 2024
%Y A216427 Cf. A143610, A216426.
%Y A216427 Cf. A001694, A002117, A004709, A013661, A013664, A062838, A085966.
%K A216427 nonn
%O A216427 1,1
%A A216427 _V. Raman_, Sep 07 2012