A309096 Increasing positive integers with prime factorization exponents all appearing earlier in the sequence.

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

1,2

Comments

Because non-existing primes in the a factorization are recorded with exponent 0 here, and because 0 is not in the sequence, all entries must have a full set of prime divisors from 2 up to their largest prime: this is a subsequence of A055932. - R. J. Mathar, May 05 2023

3 and 5 do not appear in the sequence, so entries of A176297 or A362831 are not in the sequence. - R. J. Mathar, May 05 2023

Links

Table of n, a(n) for n=1..51.

Formula

a(1) = 1; a(n) = least positive integer x > a(n-1) where the exponents e in the prime factorization of x are in a(1..n-1).

Example

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].

Prog

(Haskell)
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..]

Crossrefs

Sequence in context: A341663 A050584 A260698 * A019280 A090748 A188047

Adjacent sequences: A309093 A309094 A309095 * A309097 A309098 A309099

Keyword

nonn

Author

Chris Murray, Jul 12 2019

Status

approved