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A327314
Rectangular array read by descending antidiagonals: the distinct rows of the quotient array, A307693, of A003188.
1
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
1,2
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.
EXAMPLE
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
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 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Nov 01 2019
STATUS
approved