A386762 Perfect powers of nonsquarefree numbers k that are not squareful.

144, 324, 400, 576, 784, 1600, 1728, 1936, 2025, 2304, 2500, 2704, 2916, 3136, 3600, 3969, 4624, 5625, 5776, 5832, 6400, 7056, 7744, 8000, 8100, 8464, 9216, 9604, 9801, 10816, 12544, 13456, 13689, 13824, 14400, 15376, 15876, 17424, 18225, 18496, 19600, 20736, 21609

Offset: 1

Author

Michael De Vlieger, Aug 02 2025

Keywords

• nonn
• easy

Comments

A131605 is the union of this sequence, A303606, and A383394, where the three sequences do not intersect one another.

A001597 is the union of A131605 and A246547.

Superset of A368508 (i.e., perfect powers of superprimorials that are not powers of 2).

Examples

			Table of n, a(n) for n = 1..12:
 n    a(n)
-----------------------------
 1    144 = 12^2 = 2^4 *  3^2
 2    324 = 18^2 = 2^2 *  3^4
 3    400 = 20^2 = 2^4 *  5^2
 4    576 = 24^2 = 2^6 *  3^2
 5    784 = 28^2 = 2^4 *  7^2
 6   1600 = 40^2 = 2^6 *  5^2
 7   1728 = 12^3 = 2^6 *  3^3
 8   1936 = 44^2 = 2^4 * 11^2
 9   2025 = 45^2 = 3^4 *  5^2
10   2304 = 48^2 = 2^8 *  3^2
11   2500 = 50^2 = 2^2 *  5^4
12   2704 = 52^2 = 2^4 * 13^2
		

Links

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

Crossrefs

Cf. A001597, A126706, A131605, A246547, A286708, A303606, A368508, A383394.

Programs

Mathematica

nn = 2^15; i = 1; k = 2; MapIndexed[Set[S[First[#2]], #1] &, Select[Range@ Sqrt[nn], 1 == Min[#] < Max[#] &@ FactorInteger[#][[All, -1]] &] ]; Union@ Reap[While[j = 2; While[S[i]^j < nn, Sow[S[i]^j]; j++]; j > 2, k++; i++] ][[-1, 1]]

Python

from math import isqrt
from sympy import mobius, integer_nthroot
def A386762(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, j = 1+x-squarefreepi(integer_nthroot(x,3)[0])-squarefreepi(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()))
    return bisection(f,n,n) # Chai Wah Wu, Aug 11 2025