A126708 -id:A126708 - cp's OEIS frontend

A372404 Powerful k that are not prime powers such that k/rad(k) is nonsquarefree, where rad = A007947.

72, 108, 144, 200, 216, 288, 324, 392, 400, 432, 500, 576, 648, 675, 784, 800, 864, 968, 972, 1000, 1125, 1152, 1296, 1323, 1352, 1372, 1568, 1600, 1728, 1800, 1936, 1944, 2000, 2025, 2304, 2312, 2500, 2592, 2700, 2704, 2744, 2888, 2916, 3087, 3136, 3200, 3267, 3375, 3456

Offset: 1

Michael De Vlieger, Jun 04 2024

A001694 \ A246547 = A286708, i.e., A286708 contains powerful numbers without perfect prime powers. Hence, this sequence is a proper subset of A286708 which in turn is contained in A126706.

Numbers k in A286708 are such that rad(k)^2 | k. Numbers in this sequence are such that k != A120944(m)^2 for some m, where A120944 is the sequence of squarefree composites.

			The number 36 is not in the sequence since 36/rad(36) = 36/6 = 6, squarefree.
a(1) = 72 since 72/rad(72) = 72/6 = 12 is nonsquarefree.
a(2) = 108 since 108/rad(108) = 108/6 = 18 is nonsquarefree.
a(4) = 200 since 200/rad(200) = 200/10 = 20 is nonsquarefree, etc.

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

Cf. A001694, A007947, A013929, A120944, A126708, A177492, A126708, A246547, A286708, A332785.

With[{nn = 3300},
  Select[
    Select[Rest@ Union@ Flatten@
      Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}],
    Not@*PrimePowerQ],
  Not@ SquareFreeQ[#/(Times @@ FactorInteger[#][[;;, 1]])] &] ]

rad(n) = factorback(factorint(n)[, 1]);
isok(k) = ispowerful(k) && !isprimepower(k) && !issquarefree(k/rad(k)); \\ Michel Marcus, Jun 05 2024

A286708 = union of A177492 and this sequence.
A001694 = union of A246547, A177492, and this sequence.
A126706 = union of A332785, A177492, and this sequence.

A383394 Perfect powers of Achilles numbers.

Original entry on oeis.org

5184, 11664, 40000, 82944, 153664, 186624, 250000, 373248, 419904, 455625, 640000, 746496, 937024, 944784, 1259712, 1265625, 1327104, 1750329, 1827904, 1882384, 2458624, 3240000, 3779136, 4000000, 5345344, 6718464, 7290000, 8000000, 8340544, 9529569, 10240000

Offset: 1

Michael De Vlieger, Aug 01 2025

Proper subset of A131605, where A286708 is the union of A131605 and A052486. Therefore these are both powerful numbers and perfect powers, unlike Achilles numbers themselves.
Proper subset of A366854.
This sequence does not intersect A303606, also a proper subset of A131605.

			Table of n, a(n) for n = 1..12:
 n      a(n)
--------------------------------
 1     5184 =  72^2 = 2^6  * 3^4
 2    11664 = 108^2 = 2^4  * 3^6
 3    40000 = 200^2 = 2^6  * 5^4
 4    82944 = 288^2 = 2^10 * 3^4
 5   153664 = 392^2 = 2^6  * 7^4
 6   186624 = 432^2 = 2^8  * 3^6
 7   250000 = 500^2 = 2^4  * 5^6
 8   373248 =  72^3 = 2^9  * 3^6
 9   419904 = 648^2 = 2^6  * 3^8
10   455625 = 675^2 = 3^6  * 5^4
11   640000 = 800^2 = 2^10 * 5^4
12   746496 = 864^2 = 2^10 * 3^6

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

Cf. A001597, A052486, A126708, A131605, A286708, A303606, A366854, A386762.

nn = 2^24; mm = Sqrt[nn]; i = 1; k = 2; MapIndexed[Set[S[First[#2]], #1] &, Rest@ Select[Union@ Flatten@ Table[a^2*b^3, {b, Surd[mm, 3]}, {a, Sqrt[mm/b^3]}], GCD @@ FactorInteger[#][[;; , -1]] == 1 &]]; Union@ Reap[While[j = 2; While[S[i]^j < nn, Sow[S[i]^j]; j++]; j > 2, k++; i++] ][[-1, 1]]

from math import isqrt
from sympy import integer_nthroot, mobius
def A383394(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
        while f(kmin) < kmin: kmin >>= 1		
        kmin = max(kmin,kmax >> 1)
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    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()))-1, 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): return n+x-sum(g(integer_nthroot(x, k)[0]) for k in range(2, x.bit_length()))
    return bisection(f,n,n) # Chai Wah Wu, Aug 11 2025

cp's OEIS Frontend

A372404 Powerful k that are not prime powers such that k/rad(k) is nonsquarefree, where rad = A007947.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula