A354271 - OEIS

%I #50 May 25 2022 13:00:12

%S 2,3,2,5,7,3,11,2,5,13,17,3,11,19,2,7,23,17,29,5,13,31,7,37,3,11,19,

%T 29,41,2,5,13,43,17,47,3,11,23,41,53,2,7,37,59,31,61,5,13,23,43,67,7,

%U 19,29,37,59,71,3,11,41,73,2,5,17,47,79,19,31,53,71,83,3,67

%N Irregular array of the prime numbers read by rows.

%C The construction of the array is made in an orthogonal grid with columns and rows.

%C Along the sloping upper boundary of the array are written the guiding prime numbers, each in a column of its value and in a row of its index. From these leading entries, on downward and leftward running antidiagonal lines the preceding primes are entered. In reverse order and with the due prime gaps, these will fall into the columns of their own value, below the guiding primes on top.

%C The antidiagonals are the same as the rows of the triangle in A037126.

%C The rows that begin with 2's end with A256491.

%C Row n lists all primes of the form A000040(n - k) - k for positive k. - _Thomas Scheuerle_, May 23 2022

%H Michael De Vlieger, <a href="/A354271/b354271.txt">Table of n, a(n) for n = 1..10701</a> (rows n = 1..400, flattened)

%H Michael De Vlieger, <a href="/A354271/a354271_2.png">Bitmap</a> of 2^(pi(T(n,k)) - 1) for n = 1..1024.

%H Thomas Scheuerle, <a href="/A354271/a354271.png">The first 1000 rows, a(1 .. 58521) as scatter plot</a>.

%H Thomas Scheuerle, <a href="/A354271/a354271_1.png">Lengths of the first 10000 rows as plot against row number</a>.

%e .  2

%e .  .  3

%e .  2  .  .  5

%e .  .  .  .  .  .  7

%e .  .  3  .  .  .  .  .  .  . 11

%e .  2  .  .  5  .  .  .  .  .  .  . 13

%e .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 17

%e .  .  3  .  .  .  .  .  .  . 11  .  .  .  .  .  .  . 19

%e .  2  .  .  .  .  7  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 23

%e .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 17  .  .  .  .  .  .

%e .  .  .  .  5  .  .  .  .  .  .  . 13  .  .  .  .  .  .  .  .  .  .

%e .  .  .  .  .  .  7  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .

%e .  .  3  .  .  .  .  .  .  . 11  .  .  .  .  .  .  . 19  .  .  .  .

%e .  2  .  .  5  .  .  .  .  .  .  . 13  .  .  .  .  .  .  .  .  .  .

%e .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 17  .  .  .  .  .  .

%e .  .  3  .  .  .  .  .  .  . 11  .  .  .  .  .  .  .  .  .  .  . 23

%e .  2  .  .  .  .  7  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .

%t Table[Select[Array[Prime[#] - (n - #) &, n], And[# > 0, PrimeQ[#]] &], {n, 24}] // Flatten (* _Michael De Vlieger_, May 25 2022 *)

%t (* Extract data from the bitmap: set k to number of rows desired, up to 1024 *)

%t k = 120; Map[Prime /@ Position[#, 0.][[All, 1]] &, ImageData[Import["https://oeis.org/A354271/a354271_2.png"]][[1 ;; k]] ] // Flatten (* _Michael De Vlieger_, May 25 2022 *)

%o (MATLAB)

%o function a = A354271( max_row )

%o     p = primes(max_row*floor(2*max_row*log(max_row)));

%o     a = [];

%o     for r = 1:max_row

%o         row = p(1:r)-(r-1:-1:0);

%o         row = row(isprime(max(row,0)) > 0);

%o         a = [a row];

%o     end

%o end % _Thomas Scheuerle_, May 23 2022

%Y Cf. A000040, A037126, A256491.

%K nonn,tabf

%O 1,1

%A _Tamas Sandor Nagy_, May 22 2022