# [seqfan] 2D Flavius's sieve

Elijah Beregovsky elijah.beregovsky at gmail.com
Wed Nov 25 16:23:11 CET 2020

```I've come up with three new sequences akin to the Flavius's sieve A000960
<https://oeis.org/A000960>, but in 2D.
First, we number an infinite board diagonally (as in A316588
<https://oeis.org/A316588>). Then we delete every second diagonal. We're
left with {1,4,5,6,11,12,13,14,15,22,23...}. Then we renumber the board
with the numbers that survived the previous step and delete every third
diagonal. We're left with {1,4,5,13,14,15,22,23,24,25,26,27,42,43...}. Then
renumber again and delete every fourth diagonal, and so on. What's left is
our sequence. Another version deletes rows and another one deletes columns.
Here's the mathematica code:

dynP[l_, p_] :=
MapThread[l[[# ;; #2]] &, {{0}~Join~Most@# + 1, #} &@Accumulate at p];
Trngl := dynP[#, Range[(-1 + Sqrt[1 + 8*Length[#]])/2]] &;
lst = Trngl[Range[100]]; i = 2;
While[Length[lst] > i,
lst = Trngl[Flatten[Drop[lst, {i, -1, i}]]];
i++];
lst = Flatten[lst]

It returns {1, 4, 5, 13, 14, 15, 26, 27, 42, 43, 44, 45, 56, 57, 58}.

To get the other two sequences we change lst = Trngl[Flatten[Drop[lst, {i,
-1, i}]]]; to

lst = Trngl[Flatten[DeleteCases[Drop[PadRight[lst], None, {i, -1, i}], 0,
Infinity]]]

to yield {1, 2, 4, 6, 7, 11, 13, 16, 18, 24, 26, 28, 33, 35, 41} and to

lst = Trngl[Flatten[DeleteCases[Drop[PadLeft[lst], None, {-i, 1, -i}], 0,
Infinity]]]

to yield {1, 3, 4, 8, 10, 11, 17, 19, 21, 24, 32, 36, 37, 39, 45}.

None of these sequences are in OEIS. Are they interesting? Could you find
an asymptotic formula for them? The log-log plots for the first ten million
terms look convincingly like a line with a slope of 3/2, so I conjecture
that a(n)~n^(3/2), but I can't prove it, nor have I run a regression on
these values. Would be grateful for any ideas and contributions.