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

1,1

Comments

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

Links

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

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

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]

Prog

(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

Crossrefs

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

Cf. A013662, A013664, A013670.

Sequence in context: A223508 A250810 A378900 * A320893 A276282 A137488

Adjacent sequences: A378765 A378766 A378767 * A378769 A378770 A378771

Keyword

nonn,easy

Author

Michael De Vlieger, Dec 06 2024

Status

approved