A327314 Rectangular array read by descending antidiagonals: the distinct rows of the quotient array, A307693, of A003188.

1, 3, 1, 2, 2, 1, 6, 4, 3, 1, 7, 5, 2, 2, 1, 5, 3, 5, 4, 2, 1, 4, 8, 6, 3, 5, 2, 1, 12, 9, 4, 7, 4, 4, 3, 1, 13, 10, 10, 9, 3, 3, 2, 3, 1, 15, 7, 11, 8, 9, 8, 6, 2, 2, 1, 14, 6, 12, 6, 10, 9, 7, 6, 5, 2, 1, 10, 16, 8, 5, 11, 7, 5, 5, 4, 4, 2, 1, 11, 17, 9

Offset: 1

Author

Clark Kimberling, Nov 01 2019

Keywords

• nonn
• tabl

Comments

Each row of the quotient array, A307693, occurs infinitely many times. Specifically, if p is a prime (A000040), then for every multiple m*p of p, the rows numbered m*p are identical. In the present array only the first occurrence of each row of A307693 is retained; these are the prime-numbered rows of A307693. Every row is a permutation of the positive integers, so that every positive integer occurs infinitely many times.

Examples

			Northwest corner:
  1   3   2   6   7   5   4   12  13  15
  1   2   4   5   3   8   9   10   7   6
  1   3   2   5   6   4  10   11  12   8
  1   2   4   3   7   9   8    6   5  14
  1   2   5   4   3   9  10   11   8   7
  1   2   4   3   8   9   7    6   5  15
		

Crossrefs

Cf. A000040, A003188, A307693.

Programs

Mathematica

s = Table[BitXor[n, Floor[n/2]], {n, 2000}]; (* A003188 *)
g[n_] := Flatten[Position[Mod[s, n], 0]];
u[n_] := s[[g[Prime[n]]]]/Prime[n];
Column[Table[Take[u[n], 20], {n, 1, 20}]]  (* A326925 array *)
v[n_, k_] := u[n][[k]];
Table[v[n - k + 1, k], {n, 14}, {k, n, 1, -1}] // Flatten  (* A326925 sequence *)