A361423 Start with natural numbers, for all positive integer periods p sieve out every p-th number p-1 times over.

1, 3, 9, 27, 75, 225, 651, 1947, 5661, 15753, 44497, 128325, 357339, 1025029, 2881677, 8152327, 22251081, 62981541, 175699737, 491888331, 1353494089, 3827528649, 10655040429, 29413393659, 80737582089, 226955441541, 626061311481, 1745916338341, 4826531920159, 13166998285539

Offset

1,2

Comments

Appears to grow as: a(n) ~ c n^n/(n-1)! where c is approximately 0.56...

The terms remaining after the p-th sieve-batch grow on average with slope p^(p-1)/(p-1)!.

Links

Bert Dobbelaere, Table of n, a(n) for n = 1..600

Rok Cestnik, Sieve visualization

Index entries for sequences generated by sieves.

Example

Start with naturals: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, ...
Sieve out every 1st number 0 times (do nothing)
Sieve out every 2nd number 1 times: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, ...
Sieve out every 3rd number 2 times:
   first time: 1, 3, 7, 9, 13, 15, 19, 21, 25, 27, 31, 33, 37, 39, 43, ...
   second time: 1, 3, 9, 13, 19, 21, 27, 31, 37, 39, 45, 49, 55, 57, 63, ...
Sieve out every 4th number 3 times:
   first time: 1, 3, 9, 19, 21, 27, 37, 39, 45, 55, 57, 63, 73, 75, 81, ...
   second time: 1, 3, 9, 21, 27, 37, 45, 55, 57, 73, 75, 81, 93, 99, ...
   third time: 1, 3, 9, 27, 37, 45, 57, 73, 75, 93, 99, 109, 127, 129, ...
Sieve out every 5th number 4 times:
   first time: 1, 3, 9, 27, 45, 57, 73, 75, 99, 109, 127, 129, 153, 165, ...
   second time: 1, 3, 9, 27, 57, 73, 75, 99, 127, 129, 153, 165, 189, ...
   third time: 1, 3, 9, 27, 73, 75, 99, 127, 153, 165, 189, 201, 225, ...
   fourth time: 1, 3, 9, 27, 75, 99, 127, 153, 189, 201, 225, 261, 289, ...
Sieve out every 6th number 5 times:
   ...

Prog

(Python)
def A361423(n):
    for p in range(n, 1, -1):
        for k in range(p-1):
            n += (n-1)//(p-1)
    return n
# Bert Dobbelaere, Jul 21 2023

Crossrefs

Cf. A000960 (sieve once each).

Cf. A000959, A111039, A007950, A056533, A099267, A003309.

Sequence in context: A289693 A269684 A330079 * A135415 A182897 A228734

Adjacent sequences: A361420 A361421 A361422 * A361424 A361425 A361426

Keyword

nonn

Author

Rok Cestnik, Jul 17 2023

Extensions

More terms from Bert Dobbelaere, Jul 21 2023

Status

approved