A378768 Squares of powerful numbers that are not prime powers.

1296, 5184, 10000, 11664, 20736, 38416, 40000, 46656, 50625, 82944, 104976, 153664, 160000, 186624, 194481, 234256, 250000, 331776, 419904, 455625, 456976, 614656, 640000, 746496, 810000, 937024, 944784, 1000000, 1185921, 1265625, 1327104, 1336336, 1500625, 1679616

Offset: 1

Author

Michael De Vlieger, Dec 06 2024

Keywords

• nonn
• easy

Comments

Contained in A286708, which is a proper subset of A126706.

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000290, A001694, A126706, A286708, A372695.

Cf. A013662, A013664, A013670.

Programs

Mathematica

With[{nn = 2000}, Select[Rest@ Union[Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}] ], Not@*PrimePowerQ]^2]

Python

from math import isqrt
from sympy import integer_nthroot, primepi, mobius
def A378768(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    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):
        c, l = n+x, 0
        j = isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2, 3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        c -= squarefreepi(integer_nthroot(x, 3)[0])-l
        return c+1+sum(primepi(integer_nthroot(x, k)[0]) for k in range(2, x.bit_length()))
    return bisection(f, n, n)**2 # Chai Wah Wu, Dec 08 2024

Formula

a(n) = A286708(n)^2.

Intersection of A000290 and A286708.

Intersection of A000290 and A372695.

Sum_{n>=1} 1/a(n) = zeta(4)*zeta(6)/zeta(12) - Sum_{p prime} (1/(p^4-p^2)) - 1 = 0.0013772572536044025109... . - Amiram Eldar, Dec 10 2024