%I #65 Feb 16 2024 15:22:44

%S 48,54,96,160,162,192,224,250,320,375,384,405,448,486,567,640,686,704,

%T 768,832,896,960,1029,1080,1200,1215,1250,1280,1350,1408,1440,1458,

%U 1500,1536,1620,1664,1701,1715,1792,1875,1920,2016,2058,2160,2176,2250,2268,2352

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

%C 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.

%C 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.

%C 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.

%C Since A001694 contains A001597, the above is also true for perfect powers k; A001597 is a proper subset of A341645.

%C This sequence contains "weak" k (in A052485) such that rad(k) < k/rad(k).

%C 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.

%C 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.

%H Michael De Vlieger, <a href="/A366250/b366250.txt">Table of n, a(n) for n = 1..10000</a>

%F Set difference of A341645 and A001694.

%F Intersection of A341645 and A364702 where the latter is a proper subset of A052485.

%F Sequence contains infinite intersections of A052485 and { k = m*s : s is squarefree, rad(m) | s, 1 < s < m }.

%F {a(n)} = union of { k = s*m : s > 1 is squarefree, rad(m) | s, m >= s, k is not powerful }.

%F {a(n)} = { k in A364702 : k >= rad(k)^2 }.

%e Let b(n) = A364702(n).

%e a(1) = b(1) = 48 since rad(48) < 48/rad(48), 6 < 8.

%e b(2) = 50 is not in the sequence since rad(50) > 50/rad(50), 10 > 5.

%e a(2) = b(3) = 54 since 6 < 9, etc.

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

%o (PARI) isok(m) = if (!ispowerful(m), my(d=divisors(m)); #select(x->(issquarefree(x) && (x^2>m)), d) == 0); \\ _Michel Marcus_, Feb 11 2024

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

%K nonn

%O 1,1

%A _Peter Munn_ and _Michael De Vlieger_, Feb 08 2024