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A366250
Numbers k that are not powerful and do not have a strictly superior squarefree divisor.
5
48, 54, 96, 160, 162, 192, 224, 250, 320, 375, 384, 405, 448, 486, 567, 640, 686, 704, 768, 832, 896, 960, 1029, 1080, 1200, 1215, 1250, 1280, 1350, 1408, 1440, 1458, 1500, 1536, 1620, 1664, 1701, 1715, 1792, 1875, 1920, 2016, 2058, 2160, 2176, 2250, 2268, 2352
OFFSET
1,1
COMMENTS
A number k does not have a strictly superior squarefree divisor if and only if k is at least as large as the square of rad(k), the largest squarefree divisor of k. All powerful numbers (A001694) have this property. This sequence lists the other such numbers.
Let rad(k) = A007947(k), the largest squarefree divisor, i.e., the squarefree kernel of k. A341645 lists the numbers without a strictly superior squarefree divisor.
A341645 = { k : rad(k) <= k/rad(k) } = { k : A007947(k) <= A003557(k) }, and it is evident that rad(k) <= k/rad(k) is true for powerful k, that is, k in A001694.
Since A001694 contains A001597, the above is also true for perfect powers k; A001597 is a proper subset of A341645.
This sequence contains "weak" k (in A052485) such that rad(k) < k/rad(k).
The presence of a number, k, in this sequence depends only upon A290110(k), i.e., upon the factorization pattern of its sequence of divisors as defined in A191743.
Let S = A006939 and let P = A002110. Almost all superprimorials are in this sequence: S \ {1, 2, 12, 360} is a proper subset. S(i) = S(i-1)*P(i), where S(i-1) = A003557(S(i)) and P(i) = rad(S(i)), and for i > 4, S(i-1) > P(i). Since prime(i) | S(i) but prime(i)^2 does not divide S(i), S(i) is not powerful. Corollary: almost all superprimorials are in A341645, since this sequence is a proper subset of A341645.
LINKS
FORMULA
Set difference of A341645 and A001694.
Intersection of A341645 and A364702 where the latter is a proper subset of A052485.
Sequence contains infinite intersections of A052485 and { k = m*s : s is squarefree, rad(m) | s, 1 < s < m }.
{a(n)} = union of { k = s*m : s > 1 is squarefree, rad(m) | s, m >= s, k is not powerful }.
{a(n)} = { k in A364702 : k >= rad(k)^2 }.
EXAMPLE
Let b(n) = A364702(n).
a(1) = b(1) = 48 since rad(48) < 48/rad(48), 6 < 8.
b(2) = 50 is not in the sequence since rad(50) > 50/rad(50), 10 > 5.
a(2) = b(3) = 54 since 6 < 9, etc.
MATHEMATICA
Select[Range[2, 2400], And[! AllTrue[#2[[All, -1]], # > 1 &], #1 >= Apply[Times, #2[[All, 1]]^2]] & @@ {#, FactorInteger[#]} &]
PROG
(PARI) isok(m) = if (!ispowerful(m), my(d=divisors(m)); #select(x->(issquarefree(x) && (x^2>m)), d) == 0); \\ Michel Marcus, Feb 11 2024
KEYWORD
nonn
AUTHOR
STATUS
approved