cp's OEIS Frontend

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)

The process is described in A099361. The first application of the sieve transform produces the prime numbers A000040. The second application yields sequence A099361. In principle, the sieve transform can be applied to any sequence of positive integers. For instance, the sieve transform of the positive even numbers is 2, 4, 8, 16,.... Also note that the transform can produce a finite sequence. See A100425 and A100426 for more examples.