A390126 Powers k^m, m = 2..3, of squarefree composite k.

36, 100, 196, 216, 225, 441, 484, 676, 900, 1000, 1089, 1156, 1225, 1444, 1521, 1764, 2116, 2601, 2744, 3025, 3249, 3364, 3375, 3844, 4225, 4356, 4761, 4900, 5476, 5929, 6084, 6724, 7225, 7396, 7569, 8281, 8649, 8836, 9025, 9261, 10404, 10648, 11025, 11236, 12100

Offset

1,1

Comments

Union of A177492 and A177493.

Intersection of A046100 and A303606.

A303606 is the union of this sequence and A390127, disjoint subsets.

Analogous to A168363 = {p^m : 2 <= m <= 3, prime p}, this sequence is {k^m : 2 <= m <= 4, k in A120944}.

Links

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

Index entries for sequences related to powerful numbers.

Formula

Sum_{n>=1} 1/a(n) = zeta(2)/zeta(4) + zeta(3)/zeta(6) - P(2) - P(3) - 2 = A082020 + A157289 - A085548 - A085541 - 2 = 0.074372644304814449297..., where P is the prime zeta function. - Amiram Eldar, Mar 07 2026

Example

Table of n, a(n) for n = 1..12:
 n    a(n)
-----------------------------------
 1     36 =  6^2 = 2^2 *  3^2
 2    100 = 10^2 = 2^2 *  5^2
 3    196 = 14^2 = 2^2 *  7^2
 4    216 =  6^3 = 2^3 *  3^3
 5    225 = 15^2 = 3^2 *  5^2
 6    441 = 21^2 = 3^2 *  7^2
 7    484 = 22^2 = 2^2 * 11^2
 8    676 = 26^2 = 2^2 * 13^2
 9    900 = 30^2 = 2^2 *  3^2 * 5^2
10   1000 = 10^3 = 2^3 *  5^3
11   1089 = 33^2 = 3^2 * 11^2
12   1156 = 34^2 = 2^2 * 17^2

Mathematica

nn = 15000; i = 1; MapIndexed[Set[S[First[#2]], #1] &, Select[Range@ Sqrt[nn], And[SquareFreeQ[#], CompositeQ[#]] &]]; Union@ Reap[While[j = 2; While[And[j < 4, S[i]^j < nn], Sow[S[i]^j]; j++]; j > 2, i++] ][[-1, 1]]

Prog

(Python)
from math import isqrt
from sympy import primepi, mobius, integer_nthroot
from oeis_sequences.OEISsequences import bisection
def A390126(n):
    def f(x): return n+2+x+primepi(w:=integer_nthroot(x, 3)[0])+primepi(m:=isqrt(x))-sum(mobius(k)*(w//k**2+m//k**2) for k in range(1, isqrt(w)+1))-sum(mobius(k)*(m//k**2) for k in range(isqrt(w)+1, isqrt(m)+1))
    return bisection(f, n, n) # Chai Wah Wu, Feb 04 2026

Crossrefs

Cf. A000290, A000578, A001597, A001694, A002760, A013929, A024619, A046100, A120944, A126706, A131605, A168363, A177492, A177493, A286708, A303606, A390127.

Cf. A082020, A085541, A085548, A157289.

Sequence in context: A395414 A340017 A303606 * A303661 A390956 A175391

Adjacent sequences: A390123 A390124 A390125 * A390127 A390128 A390129

Keyword

nonn,easy

Author

Michael De Vlieger, Jan 29 2026

Status

approved