A216426 Numbers of the form a^2*b^3, where a != b and a, b > 1.

72, 108, 128, 200, 256, 288, 392, 432, 500, 512, 576, 648, 675, 800, 864, 968, 972, 1125, 1152, 1323, 1352, 1372, 1568, 1600, 1728, 1800, 1944, 2000, 2048, 2187, 2304, 2312, 2592, 2700, 2888, 2916, 3087, 3136, 3200, 3267, 3456, 3528, 3872, 3888, 4000

Offset: 1

Author

V. Raman, Sep 07 2012

Keywords

• nonn

Comments

Terms of A216427 that are not 5th powers of squarefree numbers (A113850) and not 10th powers of primes (A030629). - Amiram Eldar, Feb 07 2023

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A143610.

Subsequence of A216427. - Zak Seidov, Jan 03 2014

Cf. A002117, A013661, A013663, A013664, A013668, A030629, A085966, A113850.

Programs

Mathematica

With[{upto=4000},Select[Union[Flatten[{#[[1]]^2 #[[2]]^3,#[[2]]^2 #[[1]]^3}& /@ Subsets[Range[2,Surd[upto,2]],{2}]]],#<=upto&]](* Harvey P. Dale, Jan 04 2014 *)
pMx = 25; mx = 2^3 pMx^2; t = Flatten[Table[If[a != b, a^2 b^3, 0], {a, 2, mx^(1/2)}, {b, 2, mx^(1/3)}]]; Union[Select[t, 0 < # <= mx &]] (* T. D. Noe, Jan 02 2014 *)

PARI

list(lim)=my(v=List()); for(b=2, sqrtnint(lim\4,3), for(a=2, sqrtint(lim\b^3), if(a!=b, listput(v, a^2*b^3)))); Set(v) \\ Charles R Greathouse IV, Jan 02 2014

Python

from math import isqrt
from sympy import integer_nthroot, mobius, primepi
def A216426(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        j, b, a, d = isqrt(x), integer_nthroot(x,6)[0], integer_nthroot(x,5)[0], integer_nthroot(x,10)[0]
        l, c = 0, n+x-2+primepi(b)+sum(mobius(k)*(j//k**3) for k in range(d+1, b+1))+primepi(d)+sum(mobius(k)*(a//k**2+j//k**3) for k in range(1, d+1))
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = sum(mobius(k)*((k2-1)//k**2) for k in range(1, isqrt(k2-1)+1))
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        return c+l
    return bisection(f,n,n) # Chai Wah Wu, Sep 13 2024

Formula

Sum_{n>=1} 1/a(n) = 2 + ((zeta(2)-1)*(zeta(3)-1)-1)/zeta(6) - zeta(5)/zeta(10) - P(6) - P(10) = 0.09117811499514578262..., where P(s) is the prime zeta function. - Amiram Eldar, Feb 07 2023

Extensions

Name corrected by Charles R Greathouse IV, Jan 02 2014