A391115 - OEIS

A391115

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

%I #37 Feb 15 2026 23:12:02

%S 48,80,96,112,144,160,162,176,192,208,224,240,272,288,304,320,324,336,

%T 352,368,384,400,405,416,432,448,464,480,486,496,528,544,560,567,576,

%U 592,608,624,640,648,656,672,688,704,720,736,752,768,784,800,810,816,832

%N Numbers that are neither 4-free nor 4-full.

%C Numbers that are not in either A036967 or A046100.

%C 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.

%C Proper subset of A046101.

%C 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.

%C Union of A390379, A392134, and A392360, disjoint subsets. - _Michael De Vlieger_, Feb 05 2026

%H Michael De Vlieger, <a href="/A391115/b391115.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>.

%F 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

%e Table of n, a(n) for select n:

%e n a(n)

%e -----------------------------

%e 1 48 = 2^4 * 3

%e 2 80 = 2^4 * 5

%e 3 96 = 2^5 * 3

%e 4 112 = 2^4 * 7

%e 5 144 = 2^4 * 3^2

%e 6 160 = 2^5 * 5

%e 7 162 = 2 * 3^4

%e 8 176 = 2^4 * 11

%e 9 192 = 2^6 * 3

%e 10 208 = 2^4 * 13

%e 12 240 = 2^4 * 3 * 5

%e 114 1680 = 2^4 * 3 * 5 * 7

%e From _Michael De Vlieger_, Feb 05 2026: (Start)

%e 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:

%e 4-free 4-mid* 4-full

%e A046100 This seq. A036967 Description of partition.

%e +---------+---------+---------+ -------------------------

%e {1} | {1} | - | {1} | Empty product

%e A000040 | A000040 | - | - | Prime

%e A120944 | A120944 | - | - | Squarefree and composite

%e A332785 | A392537 | A392360 | - | Neither squarefree nor squareful

%e A052486 | A375073 | A392134 | A391011 | Squareful but not perfect power

%e +---------+---------+---------+

%e A246547 | A168363 | - | A246550 | {p^m : p in A000040, m > 1}

%e A303606 | A390126 | - | A390127 | {k^m : k in A120944, m > 1}

%e A386762 | - | A390379 | A390381 | {k^m : k in A332785, m > 1}

%e A383394 | - | - | A383394 | {k^m : k in A052486, m > 1}

%e +---------+---------+---------+

%e * 4-mid = neither 4-free nor 4-full.

%e - indicates no intersection exists. (End)

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

%o (Python)

%o from math import gcd

%o from sympy import mobius, integer_nthroot, factorint

%o from oeis_sequences.OEISsequences import bisection

%o def A391115(n):

%o def f(x):

%o c = n-1+sum(mobius(k)*(x//k**4) for k in range(1, integer_nthroot(x,4)[0]+1))

%o for u in range(1,integer_nthroot(x,7)[0]+1):

%o if all(d<=1 for d in factorint(u).values()):

%o for w in range(1,integer_nthroot(a:=x//u**7,6)[0]+1):

%o if gcd(w,u)==1 and all(d<=1 for d in factorint(w).values()):

%o for y in range(1,integer_nthroot(z:=a//w**6,5)[0]+1):

%o if gcd(w,y)==1 and gcd(u,y)==1 and all(d<=1 for d in factorint(y).values()):

%o c += integer_nthroot(z//y**5,4)[0]

%o return c

%o return bisection(f,n,n) # _Chai Wah Wu_, Jan 11 2026

%o (PARI) is_A391115(n) = if(n<=1, 0, (e->(vecmin(e)<4 && vecmax(e)>3))(factor(n)[,2])); \\ _Antti Karttunen_, Jan 22 2026

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

%Y Subsequences: A391593, A392134, A392360, A390379.

%K nonn,easy

%O 1,1

%A _Michael De Vlieger_, Jan 02 2026