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

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, 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, 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

Offset

1,2

Comments

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).

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.

Links

Table of n, a(n) for n=1..100.

Formula

If z is a nonzero value at coordinates (n,t) then

n = k*(k+t) where k is a positive integer solution of k^2 + tk - n = 0;

Moreover:

z = n/k + k - 1;

n = ((z+1)^2 - t^2)/4.

Example

Array begins (zeros replaced by dots):
                  1
                2 . 2
              3 . . . 3
            4 . . 3 . . 4
          5 . . . . . . . 5
        6 . . . 4 . 4 . . . 6
      7 . . . . . . . . . . . 7
    8 . . . . 5 . . . 5 . . . . 8
  9 . . . . . . . 5 . . . . . . . 9

Mathematica

F[n_, t_] :=
  Module[{x}, x = Floor[(-t + Sqrt[t^2 + 4 n])/2]; n - x (t + x)];
T[n_, t_] := F[n - 1, t] - F[n, t] + 1;
ARow[n_] := Table[T[n, t], {t, -(n - 1), +(n - 1)}];
Table[ARow[n], {n, 1, 10}] // Flatten

Crossrefs

Cf. A288969.

Sequence in context: A053571 A307391 A129883 * A098489 A128064 A144217

Adjacent sequences: A293575 A293576 A293577 * A293579 A293580 A293581

Keyword

nonn,tabf

Author

Luc Rousseau, Oct 12 2017

Status

approved