[seqfan] Sequence proposal by John Mason (by way of moderator)

[Olivier Gerard]

From: john.mason at lispa.it

Dear Seqfans
I propose the following sequence for your consideration. As the
construction mechanism is rather arbitrary, I will quite understand if the
sequence is considered too specific for OEIS.

The sequence consists of those natural numbers that correspond to the
values of the numeric representation of free polyominoes as counted by
A000105. The representation of a polyomino is defined as follows:
1. Place the prime numbers on the Cartesian plain, putting 2 at (0,0), 3 at
(1,0), 5 at (0,1), and so on, compiling the diagonals with successive
primes so as to form something like:
13,23, ...
 5,11,19, ...
 2, 3, 7, ...
(See also A078721)

2. Place a polyomino within the plane such that at least one square is
touching each axis.

3. Rotate and reflect the polyomino, respecting step 2, such that the
product of the primes covered by the polyomino has the minimum value. This
is the representation of the polyomino.

Hence the monomino has representation 2. The domino 6. The trominoes 30 and
42.

Define the order of the sequence as follows – that all (n)ominoes come
before the (n+1)ominoes, and that within the (n)ominoes, the
representations shall be in ascending order. It is necessary to define the
order, as some (n+1)ominoes have representation less than some (n)ominoes.

The first terms of the sequence are therefore:
2, 6, 30, 42, 210, 330, 462, 714, 1155, 2310, 2730, 3570, 3990, 7854,
10626, 15015, 19635, 22134, 26565, 62238, 72105, 30030, 39270, 43890,
46410, 51870, 53130, 67830, 110670, 132090, 138138, 144210, 147630, 149226,
180642, 243474, 255255, 257070, 324786, 336490, 420546, 451605, 456918,
608685, 1067154, 1142295, 1173102, 1174173, 1616615, 1742034, 1902318,
2667885, 2676234, 3136254, 4373358, 4755795

Some obvious consequences of the construction process:
1. The number of terms with n prime factors is equal to A000105(n).

2. The first term with n prime factors is at position sum(A000105(1), ... ,
A000105(n-1))+1


john