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A293578 Triangular array read by rows. One form of sieve of Eratosthenes (see comments for construction).

%I A293578 #8 Oct 22 2017 01:44:56
%S A293578 1,2,0,2,3,0,0,0,3,4,0,0,3,0,0,4,5,0,0,0,0,0,0,0,5,6,0,0,0,4,0,4,0,0,
%T A293578 0,6,7,0,0,0,0,0,0,0,0,0,0,0,7,8,0,0,0,0,5,0,0,0,5,0,0,0,0,8,9,0,0,0,
%U A293578 0,0,0,0,5,0,0,0,0,0,0,0,9,10,0,0,0,0,0,6,0,0,0,0,0,6,0,0,0,0,0,10
%N A293578 Triangular array read by rows. One form of sieve of Eratosthenes (see comments for construction).
%C A293578 Construction: row n >= 1 contains 2n-1 values indexed from t=-(n-1) to t=+(n-1). Initialize all values to 0. For all positive integers n and all nonnegative integers u, set the value at coordinates (n, -(n-1)) + u*(n,1) to (n + u).
%C A293578 Each nonzero value in row n corresponds to a way of writing n as a product of two positive integers (see formulas). Each row starts with a nonzero value and ends with a nonzero value. A number n is a prime iff row n contains exactly two nonzero values.
%F A293578 If z is a nonzero value at coordinates (n,t) then
%F A293578 n = k*(k+t) where k is a positive integer solution of k^2 + tk - n = 0;
%F A293578 Moreover:
%F A293578 z = n/k + k - 1;
%F A293578 n = ((z+1)^2 - t^2)/4.
%e A293578 Array begins (zeros replaced by dots):
%e A293578                   1
%e A293578                 2 . 2
%e A293578               3 . . . 3
%e A293578             4 . . 3 . . 4
%e A293578           5 . . . . . . . 5
%e A293578         6 . . . 4 . 4 . . . 6
%e A293578       7 . . . . . . . . . . . 7
%e A293578     8 . . . . 5 . . . 5 . . . . 8
%e A293578   9 . . . . . . . 5 . . . . . . . 9
%t A293578 F[n_, t_] :=
%t A293578   Module[{x}, x = Floor[(-t + Sqrt[t^2 + 4 n])/2]; n - x (t + x)];
%t A293578 T[n_, t_] := F[n - 1, t] - F[n, t] + 1;
%t A293578 ARow[n_] := Table[T[n, t], {t, -(n - 1), +(n - 1)}];
%t A293578 Table[ARow[n], {n, 1, 10}] // Flatten
%Y A293578 Cf. A288969.
%K A293578 nonn,tabf
%O A293578 1,2
%A A293578 _Luc Rousseau_, Oct 12 2017