A389558 - OEIS

A389558

Powerful numbers that are neither cubefull nor powers of squarefree numbers.

72, 108, 144, 200, 288, 324, 392, 400, 500, 576, 675, 784, 800, 968, 972, 1125, 1152, 1323, 1352, 1372, 1568, 1600, 1800, 1936, 2025, 2304, 2312, 2500, 2700, 2704, 2888, 2916, 3087, 3136, 3200, 3267, 3528, 3600, 3872, 3969, 4232, 4500, 4563, 4608, 4624, 5292, 5324

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OFFSET

1,1

COMMENTS

Excludes squares k^2 of squarefree numbers k, i.e., A062503.

Intersection of A359280 and A362147 = A359280 \ A036966 = A362147 \ A062503.

Disjoint union of A389947 and A390539 = A359280 \ A389551.

A389947 is the intersection of this sequence and A000290.

LINKS

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

Index entries for sequences related to powerful numbers.

FORMULA

From Amiram Eldar, Nov 23 2025: (Start)

Equals A325240 \ A062503.

Sum_{n>=1} 1/a(n) = A082695 - A065483 - A082020 + 1 = 0.083994528611345386799... . (End)

EXAMPLE

Let s = A390539 and let t = A389947.

Table of n, a(n) for select n:

n a(n)

-----------------------------------

1 72 2^3 * 3^2 = s(1)

2 108 2^2 * 3^3 = s(2)

3 144 2^4 * 3^2 = 12^2 = t(1)

4 200 2^3 * 5^2 = s(3)

5 288 2^5 * 3^2 = s(4)

6 324 2^2 * 3^4 = 18^2 = t(2)

7 392 2^3 * 7^2 = s(5)

8 400 2^4 * 5^2 = 20^2 = t(3)

9 500 2^2 * 5^3 = s(6)

10 576 2^6 * 3^2 = 24^2 = t(4)

11 675 3^3 * 5^2 = s(7)

23 1800 = 2^3 * 3^2 * 5^2 = s(17)

MATHEMATICA

With[{nn = 5400}, Select[Rest@ Union@ Flatten@ Table[a^2 * b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}], And[Length[#] > 1, CountDistinct[#] > 1, AnyTrue[#, # < 3 &] ] &[FactorInteger[#][[;; , -1]] ] &] ]

PROG

(Python)

from math import isqrt, gcd

from sympy import integer_nthroot, factorint

from oeis_sequences.OEISsequences import bisection, squarefreepi

def A389558(n):

def f(x):

c, l = n+x-1, 0

j = z = 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 += l-squarefreepi(integer_nthroot(x, 3)[0])-sum(isqrt(z//k**3) for k in range(1, integer_nthroot(z, 3)[0]+1) if all(d<=1 for d in factorint(k).values()))

j, l = isqrt(z), 0

while j>1:

k2 = integer_nthroot(z//j**2, 3)[0]+1

w = squarefreepi(k2-1)

c += j*(w-l)

l, j = w, isqrt(z//k2**3)

c += -l+squarefreepi(integer_nthroot(z, 3)[0])+squarefreepi(z)

for w in range(1, integer_nthroot(x, 5)[0]+1):

if all(d<=1 for d in factorint(w).values()):

for y in range(1, integer_nthroot(z:=x//w**5, 4)[0]+1):

if gcd(w, y)==1 and all(d<=1 for d in factorint(y).values()):

c += integer_nthroot(z//y**4, 3)[0]