A380857 Squares of numbers that are neither squarefree nor prime powers.

144, 324, 400, 576, 784, 1296, 1600, 1936, 2025, 2304, 2500, 2704, 2916, 3136, 3600, 3969, 4624, 5184, 5625, 5776, 6400, 7056, 7744, 8100, 8464, 9216, 9604, 9801, 10000, 10816, 11664, 12544, 13456, 13689, 14400, 15376, 15876, 17424, 18225, 18496, 19600, 20736

Offset

1,1

Comments

Proper subset of A359280 which is a proper subset of A286708 (powerful numbers that are not prime powers, a proper subset of A126706).

Does not intersect A362605.

Links

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

Formula

a(n) = A126706(n)^2.

Sum_{n>=1} 1/a(n) = Pi^2/6 - 15/Pi^2 - Sum_{p prime} 1/(p^2*(p^2-1)) = A013661 - A082020 + A085548 - A154945 = 0.025670434597226178881... . - Amiram Eldar, Feb 08 2025

Mathematica

Select[Range[150], Nor[PrimePowerQ[#], SquareFreeQ[#]] &]^2

Prog

(PARI) isok(k) = !issquarefree(k) && !isprimepower(k); \\ A126706
apply(sqr, select(isok, [1..200])) \\ Michel Marcus, Feb 07 2025
(Python)
from math import isqrt
from sympy import primepi, integer_nthroot, mobius
def A380857(n):
    def bisection(f, kmin=0, kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+sum(primepi(integer_nthroot(x, k)[0]) for k in range(2, x.bit_length()))+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))
    return bisection(f, n, n)**2 # Chai Wah Wu, Feb 08 2025

Crossrefs

Cf. A059404, A126706, A177492 (k^2 for k in A120944), A286708, A359280, A362605, A378768 (k^2 for k in A286708).

Cf. A013661, A082020, A085548, A154945.

Sequence in context: A154051 A335543 A366854 * A217584 A030633 A189988

Adjacent sequences: A380854 A380855 A380856 * A380858 A380859 A380860

Keyword

nonn,easy

Author

Michael De Vlieger, Feb 06 2025

Status

approved