A390127 Powers k^m, m >= 4, of squarefree composite k.

1296, 7776, 10000, 38416, 46656, 50625, 100000, 194481, 234256, 279936, 456976, 537824, 759375, 810000, 1000000, 1185921, 1336336, 1500625, 1679616, 2085136, 2313441, 3111696, 4084101, 4477456, 5153632, 6765201, 7529536, 9150625, 10000000, 10077696, 10556001, 11316496

Offset

1,1

Comments

Intersection of A036967 and A303606 = A388304 \ A177493.

A303606 is the union of this sequence and A390126, disjoint subsets.

This sequence is {k^m : 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) = Sum_{m>=5} (zeta(m)/zeta(2*m) - P(m) - 1) = 0.0011045546102740555225..., where P is the prime zeta function. - Amiram Eldar, Mar 07 2026

Example

Table of n, a(n) for select n:
 n    a(n)
-------------------------------------
 1     1296 =  6^4 = 2^4 *  3^4
 2     7776 =  6^5 = 2^5 *  3^5
 3    10000 = 10^4 = 2^4 *  5^4
 4    38416 = 14^4 = 2^4 *  7^4
 5    46656 =  6^6 = 2^6 *  3^6
 6    50625 = 15^4 = 3^4 *  5^4
 7   100000 = 10^5 = 2^5 *  5^5
 8   194481 = 21^4 = 3^4 *  7^4
 9   234256 = 22^4 = 2^4 * 11^4
10   279936 =  6^7 = 2^7 *  3^7
11   456976 = 26^4 = 2^4 * 13^4
14   810000 = 30^4 = 2^4 *  3^4 * 5^4

Mathematica

nn = 2^24; i = 1; MapIndexed[Set[S[First[#2]], #1] &, Select[Range@ Surd[nn, 4], And[SquareFreeQ[#], CompositeQ[#]] &]]; Union@ Reap[While[j = 4; While[S[i]^j < nn, Sow[S[i]^j]; j++]; j > 4, i++] ][[-1, 1]]

Prog

(Python)
from math import isqrt
from sympy import mobius, primepi, integer_nthroot
from oeis_sequences.OEISsequences import bisection
def A390127(n):
    def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))-primepi(x))
    def f(x): return n-4+x+(y:=x.bit_length())-sum(g(integer_nthroot(x, k)[0]) for k in range(4, y))
    return bisection(f, n+1000, n+1000) # Chai Wah Wu, Feb 04 2026

Crossrefs

Cf. A001597, A001694, A013929, A024619, A036967, A126706, A131605, A286708, A303606, A388304, A390126.

Sequence in context: A378768 A386805 A320893 * A276282 A389776 A137488

Adjacent sequences: A390124 A390125 A390126 * A390128 A390129 A390130

Keyword

nonn,easy

Author

Michael De Vlieger, Jan 29 2026

Status

approved