A288969 -id:A288969 - cp's OEIS frontend

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

Luc Rousseau, Oct 12 2017

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.

			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

Cf. A288969.

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

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.

A293497 Triangular array read by rows: row n >= 1 is the list of integers from 0 to 2n-1.

Original entry on oeis.org

0, 1, 0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8

Offset: 0

Luc Rousseau, Oct 10 2017

a(n) = the least nonnegative n - 2 * T, where T is a triangular number.
a(n) = the least nonnegative n - k * (k + 1), where k is a nonnegative integer.
This sequence shares several properties with A053186 (square excess of n):
- same recursion formula a(n) = f(n,1) with f(n,m) = if n < m then n, otherwise f(n-m,m+2);
- same formula pattern a(n) = n - g(floor(f(n))), with f and g each other's inverse function: f(x)=sqrt(x) and g(x)=x^2 in the case of A053186, f(x)=(sqrt(1+4x)-1)/2 and g(x)=x(x+1) in the case of this sequence;
- similar graphic representation (arithmetically increasing sawtooth shape);
- both sequences appear to intertwine into A288969.
Odd-indexed rows of A002262. - Omar E. Pol, Oct 10 2017

			Triangle begins:
0, 1;
0, 1, 2, 3;
0, 1, 2, 3, 4, 5;
0, 1, 2, 3, 4, 5, 6, 7;
0, 1, 2, 3, 4, 5, 6, 7, 8, 9;
...

Cf. A002262, A053186, A288969.

FOdd[x_] := x*(x + 1)
InvFOdd[x_] := (Sqrt[1 + 4 x] - 1)/2
GOdd[n_] := n - FOdd[Floor[InvFOdd[n]]]
Table[GOdd[n], {n, 0, 80}]

a(n) = n - g(floor(f(n))), with f(x) = (sqrt(1+4x)-1)/2 and g(x) = x(x+1).
a(n) = f(n,1) with f(n,m) = if n < m then n, otherwise f(n-m,m+2).
a(n) = t - t^2 + n, where t = floor(sqrt(n+1) + 1/2). - Ridouane Oudra, May 03 2019

cp's OEIS Frontend

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

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