A388549 Perfect powers k^m, m > 2, of nonsquarefree numbers k that are not squareful.

1728, 5832, 8000, 13824, 20736, 21952, 64000, 85184, 91125, 104976, 110592, 125000, 140608, 157464, 160000, 175616, 216000, 248832, 250047, 314432, 331776, 421875, 438976, 512000, 592704, 614656, 681472, 729000, 778688, 884736, 941192, 970299, 1124864, 1404928

Offset

1,1

Comments

A372695 is the union of this sequence, A383394, A388293, and A388304.

A036966 = A246549 U A372695, where A246549 is the intersection of A036966 and A000961, and A372695 is the intersection of A036966 and A024619.

Links

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

Formula

Intersection of A036966 and A386762.

Example

 n      a(n)
------------------------
 1     1728 = 2^6 *  3^3
 2     5832 = 2^3 *  3^6
 3     8000 = 2^6 *  5^3
 4    13824 = 2^9 *  3^3
 5    20736 = 2^8 *  3^4
 6    21952 = 2^6 *  7^3
 7    64000 = 2^9 *  5^3
 8    85184 = 2^6 * 11^3
 9    91125 = 3^6 *  5^3
17   216000 = 2^6 *  3^3 * 5^3
25   592704 = 2^6 *  3^3 * 7^3
28   729000 = 2^3 *  3^6 * 5^3

Mathematica

nn = 2^21; i = 1; k = 2; MapIndexed[Set[S[First[#2]], #1] &, Select[Range@ Sqrt[nn], 1 == Min[#] < Max[#] &@ FactorInteger[#][[All, -1]] &] ]; Union@ Reap[While[j = 3; While[S[i]^j < nn, Sow[S[i]^j]; j++]; j > 3, k++; i++] ][[-1, 1]]

Prog

(Python)
from math import isqrt
from sympy import integer_nthroot, mobius, factorint
from oeis_sequences.OEISsequences import bisection, squarefreepi
def A388549(n):
    def g(x):
        c, l = squarefreepi(integer_nthroot(x, 3)[0])+sum(mobius(k)*(integer_nthroot(x, k)[0]-1) for k in range(2, x.bit_length()))-2, 0
        j = 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)
        return c-l
    def f(x):
        z = isqrt(x)
        return n+2+x+g(z)+sum(mobius(k)*((w:=integer_nthroot(x, k)[0])-1)+squarefreepi(w)+g(w) for k in range(3, x.bit_length()))-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()))
    return bisection(f, n, n) # Chai Wah Wu, Dec 02 2025

Crossrefs

Cf. A036966, A246549, A332785, A383394, A386762, A388293, A388304.

Sequence in context: A202458 A387257 A392098 * A392564 A179694 A202200

Adjacent sequences: A388546 A388547 A388548 * A388550 A388551 A388552

Keyword

nonn,easy

Author

Michael De Vlieger, Sep 21 2025

Status

approved