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A293578 Triangular array read by rows. One form of sieve of Eratosthenes (see comments for construction). 3
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified June 24 08:51 EDT 2021. Contains 345416 sequences. (Running on oeis4.)