A066680 -id:A066680 - cp's OEIS frontend

A056875 Generated by sieving the natural numbers: keep the smallest remaining number k and take out its k-th successor l as well as the l-th successor m of l, the m-th successor of m and so on. Then start again from the next remaining number.

1, 3, 5, 6, 9, 10, 11, 13, 14, 18, 19, 20, 21, 23, 24, 27, 28, 30, 33, 34, 36, 38, 40, 41, 42, 43, 44, 46, 47, 50, 51, 53, 55, 58, 59, 60, 62, 65, 68, 69, 70, 71, 73, 74, 76, 79, 80, 82, 83, 84, 85, 88, 89, 91, 92, 93, 95, 96, 97, 101, 102, 103, 105, 106, 109, 111, 113, 114

Offset: 0

Thomas Schulze (jazariel(AT)tiscalenet.it), Sep 02 2000

These numbers are homogeneously distributed with a density of approximately 0.59060.

			In the first round one starts with 1 and the numbers 2,4,8,16,... are removed leaving 1,3,5,6,7,9,10,11,12,13,14,15,17,18,19,20,... The third successor of 3 is now 7 and the 7th of 7 is 15 leaving 1,3,5,6,8,9,10,11,12,13,14,16,...

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Sieve
Wikipedia, Sieve theory
Index entries for sequences generated by sieves

Cf. A066680, A232054 (complement).

a056875 n = a056875_list !! (n-1)
a056875_list =  f [1..] where
   f zs = head zs : f (g zs) where
     g (x:xs) = us ++ g vs where (us, vs) = splitAt (x - 1) xs
-- Reinhard Zumkeller, Sep 11 2013

S = Range[200]; S0 = {}; i = 1;
While[S != S0, ii = NestWhileList[#+S[[#]] &, i+S[[i]], # <= Length[S]&]; S0 = S; S = Delete[S, List /@ Select[ii, # <= Length[S]&]]; i++];
S (* Jean-François Alcover, Dec 11 2019 *)

A066683 Number of badly sieved numbers <= n.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 5, 5, 5, 6, 7, 8, 8, 8, 8, 9, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 14, 15, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 23, 23, 23, 23, 24, 24, 25, 25, 26, 26, 26, 26

Offset: 1

Reinhard Zumkeller, Dec 31 2001

nonn

Cf. A099104, A066680, A000720.

a(n) = Sum_{k=1..n} A099104(k).

A355446 Numbers of the form p^2 * q where p and q are primes with p < q < p^2.

Original entry on oeis.org

12, 45, 63, 175, 275, 325, 425, 475, 539, 575, 637, 833, 931, 1127, 1421, 1519, 1573, 1813, 2009, 2057, 2107, 2299, 2303, 2783, 2873, 3211, 3509, 3751, 3887, 4477, 4901, 4961, 5203, 5239, 5491, 5687, 6253, 6413, 6647, 6929, 7139, 7267, 7381, 7943, 8107, 8303, 8381, 8591, 8833, 8957, 8959, 9559, 9971, 10043, 10309, 10469

Offset: 1

Antti Karttunen, Jul 02 2022

nonn

Numbers whose number of divisors of n (A000005) is equal to 3 + the number of prime factors of n (with multiplicity, A001222), and the fourth smallest divisor is a square of a prime (A001248).

			12 = 2^2 * 3 is included because 2 < 3, and of the divisors of 12, [1, 2, 3, 4, 6, 12], the fourth one (4) is a square of prime as 2^2 > 3.

Setwise difference A096156 \ A355445.
Positions of 6's in A290110 and in A300250.
Subsequence of A066680, and of A355455.
A251720 is a subsequence.
Cf. A000005, A001222, A001248, A355444 (characteristic function).

Select[Range[10^4], (f = FactorInteger[#])[[;; , 2]] == {2, 1} && f[[1, 1]]^2 > f[[2, 1]] &] (* Amiram Eldar, Jul 07 2022 *)

A355444(n) = ((numdiv(n) == (3+bigomega(n))) && issquare(divisors(n)[4]));
isA355446(n) = A355444(n);

A216240 Composite numbers arising in Eratosthenes sieve with removing the multiples of every other remaining numbers after 2 (see comment).

Original entry on oeis.org

9, 21, 33, 49, 51, 77, 87, 119, 121, 123, 141, 177, 187, 201, 203, 219, 237, 287, 289, 291, 309, 319, 327, 329, 357, 393, 413, 417, 447, 451, 469, 471, 493, 501, 511, 517, 543, 553, 573, 591, 633, 649, 669, 679, 687, 697, 721, 723, 737, 763, 771, 799, 803, 807

Offset: 1

Vladimir Shevelev, Mar 14 2013

nonn

We remove even numbers except for 2. The first two remaining numbers are 3,5. Further we remove all remaining numbers multiple of 5,except for 5. The following two remaining numbers are 7,9. Now we remove all remaining numbers multiple of 9, except for 9, etc. The sequence lists the remaining composite numbers.

Conjecture. There exists x_0 such that for every x>=x_0, the number of a(n)<=x is more than pi(x).

Peter J. C. Moses, Table of n, a(n) for n = 1..10000

Cf. A003309, A038179, A004280, A054103, A055398, A055399, A066680, A083140, A145592, A152021, A152114.

Module[{a=Insert[Range[1,1000,2], 2, 2], k=4}, While[Length[a] >= 2k, a = Flatten[{Take[a,k], Select[Take[a,-Length[a]+k], Mod[#,a[[k]]] != 0 &]}]; k+=2]; Rest[Select[a,!PrimeQ[#]&]]] (* Peter J. C. Moses, Mar 27 2013 *)

cp's OEIS Frontend

A056875 Generated by sieving the natural numbers: keep the smallest remaining number k and take out its k-th successor l as well as the l-th successor m of l, the m-th successor of m and so on. Then start again from the next remaining number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

A066683 Number of badly sieved numbers <= n.

Views

Author

Keywords

Crossrefs

Formula

A355446 Numbers of the form p^2 * q where p and q are primes with p < q < p^2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

A216240 Composite numbers arising in Eratosthenes sieve with removing the multiples of every other remaining numbers after 2 (see comment).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica