A375075
Numbers whose prime factorization exponents include at least one 1, at least one 2, at least one 3 and no other exponents.
Original entry on oeis.org
360, 504, 540, 600, 756, 792, 936, 1176, 1188, 1224, 1350, 1368, 1400, 1404, 1500, 1656, 1836, 1960, 2052, 2088, 2200, 2232, 2250, 2484, 2520, 2600, 2646, 2664, 2904, 2952, 3096, 3132, 3348, 3384, 3400, 3500, 3780, 3800, 3816, 3960, 3996, 4056, 4116, 4200, 4248, 4312, 4392, 4428
Offset: 1
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Select[Range[4500], Union[FactorInteger[#][[;; , 2]]] == {1, 2, 3} &]
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is(k) = Set(factor(k)[,2]) == [1, 2, 3];
A309096
Increasing positive integers with prime factorization exponents all appearing earlier in the sequence.
Original entry on oeis.org
1, 2, 4, 6, 12, 16, 18, 30, 36, 48, 60, 64, 90, 144, 150, 162, 180, 192, 210, 240, 300, 324, 420, 450, 576, 630, 720, 810, 900, 960, 1050, 1200, 1260, 1296, 1458, 1470, 1620, 1680, 2100, 2310, 2880, 2916, 2940, 3150, 3600, 3750, 4050, 4096, 4410, 4620, 4800
Offset: 1
a(2) = 2, since 2 = 2^1 and all {1} are in a(1..1) = [1].
a(3) != 3, since 3 = 2^0 * 3^1 and not all {0,1} are in a(1..2) = [1,2].
a(3) = 4, since 4 = 2^2 and all {2} are in a(1..2) = [1,2].
a(4) != 5, since 5 = 2^0 * 3^0 * 5^1 and not all {0,1} are in a(1..3) = [1,2,4].
a(4) = 6, since 6 = 2^1 * 3^1 and all {1} are in a(1..3) = [1,2,4].
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wheelSeeds = [2, 3, 5, 7, 11, 13]
wheelOffsets = filter (\c -> all (\s -> mod c s /= 0) wheelSeeds) [1..product wheelSeeds]
restOfWheel = (concat (map (replicate (length wheelOffsets)) (map (* (product wheelSeeds)) [1..])))
wheel = wheelSeeds ++ (tail wheelOffsets) ++ (zipWith (+) (cycle wheelOffsets) restOfWheel)
isPrime n = and [n > 1, all (\c -> mod n c /= 0) (takeWhile (\c -> c * c <= n) wheel)]
primes = filter isPrime wheel
exponents bases acc n =
if (n == 1)
then (dropWhile (== 0) acc)
else if (mod n (head bases) == 0)
then (exponents bases (((head acc) + 1) : (tail acc)) (div n (head bases)))
else (exponents (tail bases) (0 : acc) n)
a = filter (\n -> all (\e -> elem e (takeWhile (<= e) a)) (exponents primes [0] n)) [1..]
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