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A307693
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Rectangular quotient array, R, of A003188 read by descending antidiagonals; see Comments.
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2
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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