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A369374
Powerful numbers k that have a primorial kernel and more than 1 distinct prime factor.
3
36, 72, 108, 144, 216, 288, 324, 432, 576, 648, 864, 900, 972, 1152, 1296, 1728, 1800, 1944, 2304, 2592, 2700, 2916, 3456, 3600, 3888, 4500, 4608, 5184, 5400, 5832, 6912, 7200, 7776, 8100, 8748, 9000, 9216, 10368, 10800, 11664, 13500, 13824, 14400, 15552, 16200
OFFSET
1,1
COMMENTS
Numbers k such that Omega(k) > omega(k) > 1 with all prime power factors p^m for m > 1, such that squarefree kernel rad(k) is in A002110, where Omega = A001222, omega = A001221, and rad(k) = A007947(k).
Union of the product of the squares of primorials P(n)^2, n > 1, and the set of prime(n)-smooth numbers.
Superset of A364930.
Proper subset of A367268, which in turn is a proper subset of A126706.
LINKS
FORMULA
{a(n)} = { m*P(n)^2 : P(n) = Product_{j = 1..n} prime(n), rad(m) | P(n), n > 1 }.
Intersection of A286708 and A055932.
A286708 is the union of A369417 and this sequence.
EXAMPLE
This sequence is the union of the following infinite sets:
P(2)^2 * A003586 = {36, 72, 108, 144, 216, 288, 324, ...}
= { m*P(2)^2 : rad(m) | P(2) }.
P(3)^2 * A051037 = {900, 1800, 2700, 3600, 4500, 5400, ...}
= { m*P(3)^2 : rad(m) | P(3) }.
P(4)^2 * A002473 = {44100, 88200, 132300, 176400, ...}
= { m*P(4)^2 : rad(m) | P(4) }, etc.
MATHEMATICA
With[{nn = 2^14},
Select[
Select[Rest@ Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}],
Not@*PrimePowerQ],
And[EvenQ[#],
Union@ Differences@ PrimePi[FactorInteger[#][[All, 1]]] == {1}] &] ]
KEYWORD
nonn
AUTHOR
Michael De Vlieger, Jan 22 2024
STATUS
approved