A391115 Numbers that are neither 4-free nor 4-full.

48, 80, 96, 112, 144, 160, 162, 176, 192, 208, 224, 240, 272, 288, 304, 320, 324, 336, 352, 368, 384, 400, 405, 416, 432, 448, 464, 480, 486, 496, 528, 544, 560, 567, 576, 592, 608, 624, 640, 648, 656, 672, 688, 704, 720, 736, 752, 768, 784, 800, 810, 816, 832

Offset

1,1

Comments

Numbers that are not in either A036967 or A046100.

Numbers that have at least 1 prime power factor p^m such that m >= 4, and at least 1 prime power factor q^m such that m < 4.

Proper subset of A046101.

Analogous to A362148 (numbers that are neither cubefree nor cubefull) and A332785 (numbers that are neither squarefree nor squareful) and a proper subset of both.

Union of A390379, A392134, and A392360, disjoint subsets. - Michael De Vlieger, Feb 05 2026

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)^s = 1 + zeta(s) - zeta(s)/zeta(4*s) - Product_{p prime}(1 + 1/(p^(3*s)*(p^s-1))), s > 1. - Amiram Eldar, Jan 09 2026

Example

Table of n, a(n) for select n:
  n    a(n)
-----------------------------
  1     48 = 2^4 *  3
  2     80 = 2^4 *  5
  3     96 = 2^5 *  3
  4    112 = 2^4 *  7
  5    144 = 2^4 *  3^2
  6    160 = 2^5 *  5
  7    162 = 2 *    3^4
  8    176 = 2^4 * 11
  9    192 = 2^6 *  3
 10    208 = 2^4 * 13
 12    240 = 2^4 *  3 * 5
114   1680 = 2^4 *  3 * 5 * 7
From Michael De Vlieger, Feb 05 2026: (Start)
Table of intersections of 4-free (A046100), 4-full (A036967), and 4-mid A391115 with certain partitions of natural numbers according to perfect powers and their roots:
           4-free    4-mid*    4-full
          A046100  This seq.  A036967   Description of partition.
        +---------+---------+---------+ -------------------------
    {1} |   {1}   |    -    |   {1}   | Empty product
A000040 | A000040 |    -    |    -    | Prime
A120944 | A120944 |    -    |    -    | Squarefree and composite
A332785 | A392537 | A392360 |    -    | Neither squarefree nor squareful
A052486 | A375073 | A392134 | A391011 | Squareful but not perfect power
        +---------+---------+---------+
A246547 | A168363 |    -    | A246550 | {p^m : p in A000040, m > 1}
A303606 | A390126 |    -    | A390127 | {k^m : k in A120944, m > 1}
A386762 |    -    | A390379 | A390381 | {k^m : k in A332785, m > 1}
A383394 |    -    |    -    | A383394 | {k^m : k in A052486, m > 1}
        +---------+---------+---------+
        * 4-mid = neither 4-free nor 4-full.
        - indicates no intersection exists. (End)

Mathematica

Select[Range[840], And[#1 < 4, #2 > 3] & @@ MinMax@ Union[FactorInteger[#][[All, -1]] ] &]

Prog

(Python)
from math import gcd
from sympy import mobius, integer_nthroot, factorint
from oeis_sequences.OEISsequences import bisection
def A391115(n):
    def f(x):
        c = n-1+sum(mobius(k)*(x//k**4) for k in range(1, integer_nthroot(x, 4)[0]+1))
        for u in range(1, integer_nthroot(x, 7)[0]+1):
            if all(d<=1 for d in factorint(u).values()):
                for w in range(1, integer_nthroot(a:=x//u**7, 6)[0]+1):
                    if gcd(w, u)==1 and all(d<=1 for d in factorint(w).values()):
                        for y in range(1, integer_nthroot(z:=a//w**6, 5)[0]+1):
                            if gcd(w, y)==1 and gcd(u, y)==1 and all(d<=1 for d in factorint(y).values()):
                                c += integer_nthroot(z//y**5, 4)[0]
        return c
    return bisection(f, n, n) # Chai Wah Wu, Jan 11 2026
(PARI) is_A391115(n) = if(n<=1, 0, (e->(vecmin(e)<4 && vecmax(e)>3))(factor(n)[, 2])); \\ Antti Karttunen, Jan 22 2026

Crossrefs

Cf. A036967, A046100, A046101, A332785, A362148.

Subsequences: A391593, A392134, A392360, A390379.

Sequence in context: A261553 A335281 A335196 * A391591 A348527 A392360

Adjacent sequences: A391112 A391113 A391114 * A391116 A391117 A391118

Keyword

nonn,easy

Author

Michael De Vlieger, Jan 02 2026

Status

approved