A390540 Cubefull powers of squarefree numbers.

8, 16, 27, 32, 64, 81, 125, 128, 216, 243, 256, 343, 512, 625, 729, 1000, 1024, 1296, 1331, 2048, 2187, 2197, 2401, 2744, 3125, 3375, 4096, 4913, 6561, 6859, 7776, 8192, 9261, 10000, 10648, 12167, 14641, 15625, 16384, 16807, 17576, 19683, 24389, 27000, 28561, 29791

Offset

1,1

Comments

Intersection of A036966 and A072777.

A072777, the sequence squares of squarefree numbers that are not squarefree, is the union of this sequence, and A062503.

Union of A246549 (i.e., cubefull prime powers) and A388304 (cubefull numbers in A303606).

Links

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

Formula

Sum_{n>=1} 1/a(n) = A368250 - A082020 + 1 = 0.32881611332981706102... . - Amiram Eldar, Nov 11 2025

Example

Table of n, a(n) for select n:
 n     a(n)
------------------------------------
 1       8 =  2^3
 2      16 =  2^4
 3      27 =  3^3
 4      32 =  2^5
 5      64 =  2^6
 6      81 =  3^4
 7     125 =  5^3
 8     128 =  2^7
 9     216 =  6^3 = 2^3 * 3^3
16    1000 = 10^3 = 2^3 * 5^3
18    1296 =  6^4 = 2^4 * 3^4
44   27000 = 30^3 = 2^3 * 3^3 * 5^3

Maple

N:= 30000: # for terms <= N
SF:= select(numtheory:-issqrfree, [seq(i, i=2..floor(N^(1/3)))]):
sort(map(proc(t) local i;  seq(t^i, i=3..ilog[t](N)) end proc, SF)); # Robert Israel, Nov 11 2025

Mathematica

nn = 2^15; mm = Sqrt[nn]; i = 1; MapIndexed[Set[S[First[#2]], #1] &, Select[Range[2, Sqrt[nn]], SquareFreeQ] ]; Union@ Reap[ While[j = 3; While[S[i]^j < nn, Sow[S[i]^j]; j++]; j > 3, i++] ][[-1, 1]]

Prog

(Python)
from sympy import integer_nthroot
from oeis_sequences.OEISsequences import bisection, squarefreepi
def A390540(n):
    def f(x): return n+x-sum(squarefreepi(integer_nthroot(x, k)[0])-1 for k in range(3, x.bit_length()))
    return bisection(f, n, n) # Chai Wah Wu, Nov 11 2025
(PARI) isok(k) = ispower(k, , &m) && valuation(k, m) > 2 && issquarefree(m); \\ Amiram Eldar, Nov 11 2025

Crossrefs

Cf. A001597, A001694, A013929, A036966, A062503, A072777, A246547, A246549, A303606, A388304.

Cf. A082020, A368250.

Sequence in context: A036966 A076467 A111231 * A111307 A386632 A246549

Adjacent sequences: A390537 A390538 A390539 * A390541 A390542 A390543

Keyword

nonn,easy

Author

Michael De Vlieger, Nov 10 2025

Status

approved