A390905 Powers k^m, m > 2, of numbers k that are neither prime powers nor powerful.

216, 1000, 1296, 1728, 2744, 3375, 5832, 7776, 8000, 9261, 10000, 10648, 13824, 17576, 20736, 21952, 27000, 35937, 38416, 39304, 42875, 46656, 50625, 54872, 59319, 64000, 74088, 85184, 91125, 97336, 100000, 104976, 110592, 125000, 132651, 140608, 157464, 160000

Offset

1,1

Comments

Powers k^m, m > 2, of composite weak k, i.e., k is in the intersection of A002808 and A052486 = A389065.

Union of A388304 and A388549.

A390823 is the disjoint union of this sequence and A387254.

A387255 is the disjoint union of this sequence and A383394.

A076467 is the disjoint union of this sequence, A246549, and A383394.

A036966 is the disjoint union of this sequence, A246549, A383394, and A388293.

Links

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

Index entries for sequences related to powerful numbers.

Example

Let s = A388304 and let t = A388549.
Table of n, a(n) for select n:
 n     a(n)
-------------------------------------------
 1     216 =  6^3 = s(1)  = 2^3 * 3^3
 2    1000 = 10^3 = s(2)  = 2^3 * 5^3
 3    1296 =  6^4 = s(3)  = 2^4 * 3^4
 4    1728 = 12^3 = t(1)  = 2^6 * 3^3
 5    2744 = 14^3 = s(4)  = 2^3 * 7^3
 6    3375 = 15^3 = s(5)  = 3^3 * 5^3
 7    5832 = 18^3 = t(2)  = 2^3 * 3^6
 8    7776 =  6^5 = s(6)  = 2^5 * 3^5
 9    8000 = 20^3 = t(3)  = 2^6 * 5^3
10    9261 = 21^3 = s(7)  = 3^3 * 7^3
11   10000 = 10^4 = s(8)  = 2^4 * 5^4
17   27000 = 30^3 = s(11) = 2^3 * 3^3 * 5^3

Mathematica

nn = 2^41; Union@ Flatten@ Table[n = a^5*b^4*c^3; Set[{k, m, r}, {#1, #2, If[FreeQ[#3/#2, 1], GCD @@ (#3/#2), 0]} & @@ {Length[#], GCD @@ #, #}] &[FactorInteger[n][[;; , -1]] ]; If[And[k > 1, m > 1], If[r != 1, n, Nothing], Nothing], {c, Surd[nn, 3]}, {b, Surd[nn/(c^3), 4]}, {a, Surd[nn/(b^4*c^3), 5] } ]

Prog

(Python)
from math import isqrt
from sympy import integer_nthroot, primepi, factorint
from oeis_sequences.OEISsequences import bisection, squarefreepi
def A390905(n):
    def g(x):
        c, l, j = x-squarefreepi(integer_nthroot(x, 3)[0])-primepi(x), 0, isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2, 3)[0]+1
            w = squarefreepi(k2-1)
            c += j*(l-w)
            l, j = w, isqrt(x//k2**3)
        return c+l
    def f(x): return n+x-sum(g(integer_nthroot(x, k)[0]) for k in range(2, x.bit_length()))+(y:=isqrt(x))-sum(isqrt(y//k**3) for k in range(1, integer_nthroot(y, 3)[0]+1) if all(d<=1 for d in factorint(k).values()))-primepi(y)
    return bisection(f, n, n) # Chai Wah Wu, Dec 02 2025

Crossrefs

Cf. A002808, A036966, A076467, A052485, A126706, A286708, A246549, A383394, A387254, A387255, A388304, A388549, A389065, A390823.

Sequence in context: A386801 A109399 A387255 * A388304 A384006 A111029

Adjacent sequences: A390902 A390903 A390904 * A390906 A390907 A390908

Keyword

nonn,easy

Author

Michael De Vlieger, Nov 24 2025

Status

approved