A100424 -id:A100424 - cp's OEIS frontend

A152009 (L)-sieve transform of {1,4,7,10,...,3n-2,...} (A016777).

1, 3, 6, 10, 16, 25, 39, 60, 91, 138, 208, 313, 471, 708, 1063, 1596, 2395, 3594, 5392, 8089, 12135, 18204, 27307, 40962, 61444, 92167

Offset: 1

John W. Layman, Nov 19 2008

nonn

The (L)-sieve transform of the sequence {a(n)} of positive integers is defined as follows: Denote the sequence of natural numbers by N. Remove the first term of N, which we denote by s(1) and then from the resulting sequence delete all terms whose index is a term of {a(n)}, to obtain the sequence N'.
Then remove the first term of N', denoted by s(2) and then from the resulting sequence delete all terms whose index is a term of {a(n)}, to obtain N''. Repeat this process indefinitely to obtain the transform LST({a(n)}) = {s(1), s(2),...}, the sequence of initial terms removed at each stage.
The (L)-sieve transform is quite different from the transform introduced by N. J. A. Sloane in A099361 and used by T. D. Noe in A100424 - A100426 and seems to lead to more interesting results and relationships among sequences. An interesting property of the (L)-sieve transform is that the (L)-sieve transform of the sequence {1,3,6,10,...,n(n+1)/2,...} of triangular numbers is again the triangular numbers. Another (conjectured) connection with the triangular numbers is given in the following.
Conjecture. Let x(0) be a random sequence of positive integers and, for n>0, let x(n)=S[x(n-1)], where S is the (L)-sieve transform. Then the limit of {x(n)} as n goes to infinity is the sequence of triangular numbers {1,3,6,10,...,n(n+1)/2,...}.
Illustration of the conjecture:
x(0)={3,8,12,14,18,22,25,31,34,39,42,45,...} (A random initial sequence.)
x(1)={1,2,3,5,7,10,14,20,28,38,51,69,...}
x(2)={1,5,12,20,30,41,53,65,78,91,105,119,...}
x(3)={1,3,5,8,11,15,19,24,29,35,41,48,...}
x(4)={1,3,7,13,21,31,43,56,71,88,107,127,...}
x(5)={1,3,6,10,15,20,26,33,40,48,56,65,...}
x(6)={1,3,6,10,15,22,30,39,50,62,75,90,...}
x(7)={1,3,6,10,15,21,28,36,45,55,66,78,...} ...
t={1,3,6,10,15,21,28,36,45,55,66,78,...} (Triangular numbers)

a[1]:1$
a[n]:=floor((3*a[n-1]+3)/2)$
A152009(n):=a[n];
makelist(A152009(n),n,1,30); /* Martin Ettl, Oct 31 2012 */

It appears that {a(n)} is given by a(n)=floor[(3*a(n-1)+3)/2], with a(1)=1.

A099361 A variation on the sieve of Eratosthenes (A000040): Start with the primes; the first term is 2, which is a(1) and we cross off every second prime starting with 2; the next prime not crossed off is 3, which is a(2) and we cross off every third prime starting with 3; the next prime not crossed off is 7, which is a(3) and we cross off every 7th prime starting with 7; and so on.

Original entry on oeis.org

2, 3, 7, 13, 29, 37, 53, 79, 89, 107, 113, 139, 151, 173, 181, 223, 239, 251, 311, 317, 349, 359, 383, 397, 421, 463, 491, 503, 541, 577, 593, 613, 619, 647, 659, 683, 743, 787, 821, 857, 863, 887, 911, 983, 997, 1033, 1061, 1151, 1163, 1193, 1213, 1249

Offset: 1

N. J. A. Sloane, Nov 18 2004

In contrast to Flavius's sieve (A000960), primes are not erased when they are crossed off; that is, primes get crossed off multiple times (see A099362).

			The first few sieving stages are as follows (X or XX indicates a prime that has been crossed off one or more times):
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 ...
2 3 X 7 XX 13 XX 19 XX 29 XX 37 XX 43 XX 53 XX 61 XX 71 XX 79 XX 89 XX ...
2 3 X 7 XX 13 XX XX XX 29 XX 37 XX XX XX 53 XX 61 XX XX XX 79 XX 89 XX ...
2 3 X 7 XX 13 XX XX XX 29 XX 37 XX XX XX 53 XX XX XX XX XX 79 XX 89 XX ...
.... Continue forever and the numbers not crossed off give the sequence.

Cf. A000040, A000960, A099204, A099207, A099243, A099362.

Cf. A100424.

nn=300; a=Prime[Range[nn]]; Do[p=a[[i]]; If[p>0, Do[a[[j]]=0, {j, i+p, nn, p}]], {i, nn}]; Rest[Union[a]] (* T. D. Noe, Nov 18 2004 *)

More terms from T. D. Noe and Ray Chandler, Nov 18 2004

A100426 A sieve transform applied to Euler's phi function A000010 starting at 4.

Original entry on oeis.org

2, 4, 6, 10, 8, 18, 24, 40, 80, 84, 144, 176, 224, 384, 360, 464, 480, 544, 704, 864, 840, 864, 1012, 1184, 1280, 1344, 1320, 1640, 1904, 2016, 2384, 2496, 2864, 2600, 2976, 3104, 3240, 3824, 3560, 4544, 4784, 4704, 5184, 5264, 5376, 5664, 5600, 6080, 5640

Offset: 1

T. D. Noe, Nov 19 2004

nonn

Cf. A099361, A100424, A100425.

(* see A100424 for the definition of SieveTransform *) SieveTransform[EulerPhi[Range[4, 20000]]]

A100425 A sieve transform applied to the composite numbers.

Original entry on oeis.org

4, 6, 8, 9, 12, 14, 18, 21, 25, 26, 35, 36, 50, 52, 56, 66, 68, 74, 77, 95, 96, 98, 108, 118, 141, 143, 146, 162, 166, 176, 185, 186, 187, 203, 207, 228, 235, 245, 250, 254, 273, 275, 285, 290, 295, 302, 322, 340, 350, 361, 375, 380, 402, 404, 416, 417, 436, 445

Offset: 1

T. D. Noe, Nov 19 2004

nonn

Cf. A099361, A100424, A100426.

(* see A100424 for the definition of SieveTransform *) SieveTransform[Complement[Range[2, 1000], Prime[Range[PrimePi[1000]]]]]

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A152009 (L)-sieve transform of {1,4,7,10,...,3n-2,...} (A016777).

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A100426 A sieve transform applied to Euler's phi function A000010 starting at 4.

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A100425 A sieve transform applied to the composite numbers.

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