A359280 Powerful numbers that are neither prime powers nor powers of squarefree composites.

72, 108, 144, 200, 288, 324, 392, 400, 432, 500, 576, 648, 675, 784, 800, 864, 968, 972, 1125, 1152, 1323, 1352, 1372, 1568, 1600, 1728, 1800, 1936, 1944, 2000, 2025, 2304, 2312, 2500, 2592, 2700, 2704, 2888, 2916, 3087, 3136, 3200, 3267, 3456, 3528, 3600, 3872, 3888, 3969

Offset: 1

Author

Michael De Vlieger, Aug 01 2023

Keywords

• nonn

Comments

Numbers k such that omega(k) > 1 and for prime power factors p^e | k, multiplicities e > 1, yet the multiplicities are not equal.

Subset of A286708, which in turn is a subset of A361098, itself a subset of A126706, the sequence of numbers neither squarefree nor prime powers.

Since A001694 = Union({1}, A246547, A286708), this sequence is a subset of A001694.

Examples

			Let b(n) = A286708(n).
b(1) = 36 is not in the sequence since rad(36) = A007947(36) = 6, and 36 = 6^2.
b(2) = a(1) = 72 since 72 is not a perfect power of rad(72).
b(3) = 100 = rad(100)^2 = 10^2, so it is not in the sequence.
b(4) = a(2) = 108, since 108 is not a perfect power of rad(108) = 6.
b(5) = a(3) = 144, since 144 is not a perfect power of rad(144) = 6.
b(6) = 196 is not in the sequence since 196 = rad(196)^2 = 14^2, etc.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Plot A001694(ym + x) at (x, y) for n = ym + x, m = 1024 and x = 1..m, y = 0..m-1, showing terms in this sequence in black, and both prime powers and those in A303606 in white.
• Michael De Vlieger, Plot A286708(n) at (x, y) for n = ym + x, m = 1024 and x = 1..m, y = 0..m-1, showing terms in this sequence in black, and those in A303606 in white.

Crossrefs

Cf. A001694, A007947, A082695, A126706, A286708, A303606, A361098.

Programs

Mathematica

nn = 5000; s = Rest@ Select[Union@ Flatten@Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], Not@*PrimePowerQ]; Select[s, !SameQ @@ FactorInteger[#][[All, -1]] &]

Python

from math import isqrt
from sympy import mobius, integer_nthroot
def A359280(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
        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):
        j = isqrt(x)
        c, l = n+x+3-(y:=x.bit_length())+squarefreepi(j)+sum(squarefreepi(integer_nthroot(x, k)[0]) for k in range(4, y)), 0
        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)
        return c+l
    return bisection(f,n,n) # Chai Wah Wu, Feb 09 2025

Formula

This sequence is A286708 \ A303606.

Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - Sum_{k>=2} (zeta(k)/zeta(2*k) - 1) - 1 = 0.094962568855... . - Amiram Eldar, Dec 09 2023