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A354271 Irregular array of the prime numbers read by rows.

%I A354271 #50 May 25 2022 13:00:12
%S A354271 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 A354271 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 A354271 19,29,37,59,71,3,11,41,73,2,5,17,47,79,19,31,53,71,83,3,67
%N A354271 Irregular array of the prime numbers read by rows.
%C A354271 The construction of the array is made in an orthogonal grid with columns and rows.
%C A354271 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 A354271 The antidiagonals are the same as the rows of the triangle in A037126.
%C A354271 The rows that begin with 2's end with A256491.
%C A354271 Row n lists all primes of the form A000040(n - k) - k for positive k. - _Thomas Scheuerle_, May 23 2022
%H A354271 Michael De Vlieger, <a href="/A354271/b354271.txt">Table of n, a(n) for n = 1..10701</a> (rows n = 1..400, flattened)
%H A354271 Michael De Vlieger, <a href="/A354271/a354271_2.png">Bitmap</a> of 2^(pi(T(n,k)) - 1) for n = 1..1024.
%H A354271 Thomas Scheuerle, <a href="/A354271/a354271.png">The first 1000 rows, a(1 .. 58521) as scatter plot</a>.
%H A354271 Thomas Scheuerle, <a href="/A354271/a354271_1.png">Lengths of the first 10000 rows as plot against row number</a>.
%e A354271 .  2
%e A354271 .  .  3
%e A354271 .  2  .  .  5
%e A354271 .  .  .  .  .  .  7
%e A354271 .  .  3  .  .  .  .  .  .  . 11
%e A354271 .  2  .  .  5  .  .  .  .  .  .  . 13
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%e A354271 .  .  3  .  .  .  .  .  .  . 11  .  .  .  .  .  .  . 19
%e A354271 .  2  .  .  .  .  7  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 23
%e A354271 .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 17  .  .  .  .  .  .
%e A354271 .  .  .  .  5  .  .  .  .  .  .  . 13  .  .  .  .  .  .  .  .  .  .
%e A354271 .  .  .  .  .  .  7  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
%e A354271 .  .  3  .  .  .  .  .  .  . 11  .  .  .  .  .  .  . 19  .  .  .  .
%e A354271 .  2  .  .  5  .  .  .  .  .  .  . 13  .  .  .  .  .  .  .  .  .  .
%e A354271 .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 17  .  .  .  .  .  .
%e A354271 .  .  3  .  .  .  .  .  .  . 11  .  .  .  .  .  .  .  .  .  .  . 23
%e A354271 .  2  .  .  .  .  7  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
%t A354271 Table[Select[Array[Prime[#] - (n - #) &, n], And[# > 0, PrimeQ[#]] &], {n, 24}] // Flatten (* _Michael De Vlieger_, May 25 2022 *)
%t A354271 (* Extract data from the bitmap: set k to number of rows desired, up to 1024 *)
%t A354271 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 A354271 (MATLAB)
%o A354271 function a = A354271( max_row )
%o A354271     p = primes(max_row*floor(2*max_row*log(max_row)));
%o A354271     a = [];
%o A354271     for r = 1:max_row
%o A354271         row = p(1:r)-(r-1:-1:0);
%o A354271         row = row(isprime(max(row,0)) > 0);
%o A354271         a = [a row];
%o A354271     end
%o A354271 end % _Thomas Scheuerle_, May 23 2022
%Y A354271 Cf. A000040, A037126, A256491.
%K A354271 nonn,tabf
%O A354271 1,1
%A A354271 _Tamas Sandor Nagy_, May 22 2022