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

Author

Michael De Vlieger, Feb 06 2025

Keywords

• nonn
• easy

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

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.

Programs

Mathematica

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

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

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