A327314 -id:A327314 - cp's OEIS frontend

A307693 Rectangular quotient array, R, of A003188 read by descending antidiagonals; see Comments.

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

Offset: 1

Clark Kimberling, Oct 26 2019

Suppose that P = (p(m)) is a permutation of the positive integers, such as A038722. For each n >= 1, let q(n,k) be the k-th index m such that n divides p(m), and let r(n) = p(q(n,k))/n. Let R be the array having (r(n)) as row n. We call R the quotient array of P. Every row of R is a permutation of the positive integers.

In the present case that P = A003188, every row 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. See A327314 for the array that results by deleting duplicate rows from R.

			A003188 = (1, 3, 2, 6, 7, 5, 4, 12, 13, 15, 14, 10, 11, 9, 8, 24, 25, 27, 26, 30, 31, 29, 28, 20, ...)
Row 1 of R is just A003188. To get row 2 of R, skip the odds in A003188 and divide the evens by 2; row 2 equals row 1. Generally, to get row n, divide A003188 by n and then delete the non-integers.
________________
Northwest corner of R:
  1   3   2   6   7   5    4   12   13   15
  1   3   2   6   7   5    4   12   13   15
  1   2   4   5   3   8    9   10    7    6
  1   3   2   6   7   5    4   12   13   15
  1   3   2   5   6   4   10   11   12    8
  1   2   4   5   3   8    9   10    7    6

Cf. A000040, A003188, A327314.

s = Table[BitXor[n, Floor[n/2]], {n, 300}]  (* A003188 *)
g[n_] := Flatten[Position[Mod[s, n], 0]];
u[n_] := s[[g[n]]]/n;
TableForm[Table[Take[u[n], 10], {n, 1, 20}]]  (* A307693 array *)
v[n_, k_] := u[n][[k]]
Table[v[n - k + 1, k], {n, 14}, {k, n, 1, -1}] // Flatten (* A307693 sequence *)

cp's OEIS Frontend

A307693 Rectangular quotient array, R, of A003188 read by descending antidiagonals; see Comments.

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