A389947 Squares of nonsquarefree numbers k that are not squareful.

144, 324, 400, 576, 784, 1600, 1936, 2025, 2304, 2500, 2704, 2916, 3136, 3600, 3969, 4624, 5625, 5776, 6400, 7056, 7744, 8100, 8464, 9216, 9604, 9801, 10816, 12544, 13456, 13689, 14400, 15376, 15876, 17424, 18225, 18496, 19600, 21609, 21904, 22500, 23104, 23409, 24336, 25600, 26244, 26896, 28224, 29241, 29584, 30625, 30976, 32400

Offset

1,1

Comments

Numbers in A386762 that are not cubefull.

Numbers in A389864 that are not cubefull, where A389864 is the union of A386762 and A383394, since the latter is a proper subset of A036966.

Intersection of A386762 and A362147 = intersection of A389864 and A362147.

Squares of terms in A332785. - Chai Wah Wu, Dec 01 2025

Superset of A392069; a(52) = 32400 is not in A392069.

Links

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

Index entries for sequences related to powerful numbers.

Formula

From Amiram Eldar, Nov 23 2025: (Start)

Equals (A000290 \ A374291) \ A062503.

Sum_{n>=1} 1/a(n) = 1 + Pi^2/6 - 35745/(1382*Pi^2) = 0.02429317734362177637... . (End)

Example

Table of n, a(n) for select n:
 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   1936 = 44^2 = 2^4 * 11^2
 8   2025 = 45^2 = 3^4 *  5^2
 9   2304 = 48^2 = 2^8 *  3^2
10   2500 = 50^2 = 2^2 *  5^4
11   2704 = 52^2 = 2^4 * 13^2
14   3600 = 60^2 = 2^4 *  3^2 * 5^2

Mathematica

Select[Range[155], And[#1 > 1, 0 < #2 < #1] & @@ {Length[#], Count[#[[;; , -1]], 1]} &[FactorInteger[#]] &]^2

Prog

(Python)
from math import isqrt
from sympy import integer_nthroot
from oeis_sequences.OEISsequences import bisection, squarefreepi
def A389947(n):
    def f(x):
        c, l, y = n+x-1, 0, isqrt(x)
        j = isqrt(y)
        while j>1:
            k2 = integer_nthroot(y//j**2, 3)[0]+1
            w = squarefreepi(k2-1)
            c += j*(w-l)
            l, j = w, isqrt(y//k2**3)
        return c-y-l+squarefreepi(integer_nthroot(y, 3)[0])+squarefreepi(y)
    return bisection(f, n, n) # Chai Wah Wu, Dec 01 2025

Crossrefs

Cf. A000290, A001597, A001694, A013929, A024619, A036966, A062503, A126706, A131605, A286708, A332785, A362147, A374291, A383394, A386762, A389864, A392069.

Sequence in context: A386762 A390379 A392424 * A392069 A217584 A030633

Adjacent sequences: A389944 A389945 A389946 * A389948 A389949 A389950

Keyword

nonn,easy

Author

Michael De Vlieger, Nov 17 2025

Status

approved