A366250 Numbers k that are not powerful and do not have a strictly superior squarefree divisor.

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

Author

Peter Munn and Michael De Vlieger, Feb 08 2024

Keywords

• nonn

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A001694, A002110, A003557, A006939, A007947, A052485, A191743, A290110, A341645, A364702.

Programs

Mathematica

Select[Range[2, 2400], And[! AllTrue[#2[[All, -1]], # > 1 &], #1 >= Apply[Times, #2[[All, 1]]^2]] & @@ {#, FactorInteger[#]} &]

PARI

isok(m) = if (!ispowerful(m), my(d=divisors(m)); #select(x->(issquarefree(x) && (x^2>m)), d) == 0); \\ Michel Marcus, Feb 11 2024

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 }.