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

1,1

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.

Union of disjoint sequences A052486, A383394, and A386762, i.e., A059404 \ A332785. - Michael De Vlieger, Oct 04 2025

From Michael De Vlieger, Nov 28 2025: (Start)

Proper subset of A046099, since all terms are divisible by a cube.

A046099 is the disjoint union of this sequence and A390540, where A390540 is the union of A246549 (powers p^m, m > 2, of prime p) and A388304 (powers k^m, m > 2, of squarefree composite k). (End)

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.

Index entries for sequences related to powerful numbers.

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

Example

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.

Mathematica

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

Prog

(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

Crossrefs

Cf. A001694, A007947, A046099, A052486, A059404, A082695, A126706, A286708, A303606, A332785, A361098, A383394, A386762, A390540.

Sequence in context: A372404 A391921 A356871 * A395067 A389558 A391491

Adjacent sequences: A359277 A359278 A359279 * A359281 A359282 A359283

Keyword

nonn

Author

Michael De Vlieger, Aug 01 2023

Status

approved