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

%I #178 Jul 23 2023 02:08:03

%S 1,3,9,27,75,225,651,1947,5661,15753,44497,128325,357339,1025029,

%T 2881677,8152327,22251081,62981541,175699737,491888331,1353494089,

%U 3827528649,10655040429,29413393659,80737582089,226955441541,626061311481,1745916338341,4826531920159,13166998285539

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

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

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

%H Bert Dobbelaere, <a href="/A361423/b361423.txt">Table of n, a(n) for n = 1..600</a>

%H Rok Cestnik, <a href="/A361423/a361423.pdf">Sieve visualization</a>

%H <a href="https://oeis.org/index/Si#sieve">Index entries for sequences generated by sieves</a>.

%e Start with naturals: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, ...

%e Sieve out every 1st number 0 times (do nothing)

%e Sieve out every 2nd number 1 times: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, ...

%e Sieve out every 3rd number 2 times:

%e first time: 1, 3, 7, 9, 13, 15, 19, 21, 25, 27, 31, 33, 37, 39, 43, ...

%e second time: 1, 3, 9, 13, 19, 21, 27, 31, 37, 39, 45, 49, 55, 57, 63, ...

%e Sieve out every 4th number 3 times:

%e first time: 1, 3, 9, 19, 21, 27, 37, 39, 45, 55, 57, 63, 73, 75, 81, ...

%e second time: 1, 3, 9, 21, 27, 37, 45, 55, 57, 73, 75, 81, 93, 99, ...

%e third time: 1, 3, 9, 27, 37, 45, 57, 73, 75, 93, 99, 109, 127, 129, ...

%e Sieve out every 5th number 4 times:

%e first time: 1, 3, 9, 27, 45, 57, 73, 75, 99, 109, 127, 129, 153, 165, ...

%e second time: 1, 3, 9, 27, 57, 73, 75, 99, 127, 129, 153, 165, 189, ...

%e third time: 1, 3, 9, 27, 73, 75, 99, 127, 153, 165, 189, 201, 225, ...

%e fourth time: 1, 3, 9, 27, 75, 99, 127, 153, 189, 201, 225, 261, 289, ...

%e Sieve out every 6th number 5 times:

%e ...

%o (Python)

%o def A361423(n):

%o for p in range(n,1,-1):

%o for k in range(p-1):

%o n += (n-1)//(p-1)

%o return n

%o # _Bert Dobbelaere_, Jul 21 2023

%Y Cf. A000960 (sieve once each).

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

%K nonn

%O 1,2

%A _Rok Cestnik_, Jul 17 2023

%E More terms from _Bert Dobbelaere_, Jul 21 2023