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A372619
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Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = 1/(phi(k)) * Sum_{j=1..n} phi(k*j).
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9
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1, 1, 2, 1, 3, 4, 1, 2, 5, 6, 1, 3, 5, 9, 10, 1, 2, 5, 7, 13, 12, 1, 3, 4, 9, 11, 17, 18, 1, 2, 6, 6, 13, 14, 23, 22, 1, 3, 4, 10, 11, 17, 20, 31, 28, 1, 2, 5, 6, 14, 13, 23, 24, 37, 32, 1, 3, 5, 9, 10, 20, 19, 31, 33, 45, 42, 1, 2, 5, 7, 13, 12, 26, 23, 37, 37, 55, 46
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OFFSET
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1,3
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LINKS
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FORMULA
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T(n,k) ~ (3/Pi^2) * c(k) * n^2, where c(k) = A078615(k)/A322360(k) is the multiplicative function defined by c(p^e) = p^2/(p^2-1). - Amiram Eldar, May 09 2024
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EXAMPLE
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Square array T(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
2, 3, 2, 3, 2, 3, 2, 3, 2, 3, ...
4, 5, 5, 5, 4, 6, 4, 5, 5, 5, ...
6, 9, 7, 9, 6, 10, 6, 9, 7, 9, ...
10, 13, 11, 13, 11, 14, 10, 13, 11, 14, ...
12, 17, 14, 17, 13, 20, 12, 17, 14, 18, ...
18, 23, 20, 23, 19, 26, 19, 23, 20, 24, ...
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MATHEMATICA
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T[n_, k_] := Sum[EulerPhi[k*j], {j, 1, n}] / EulerPhi[k]; Table[T[k, n-k+1], {n, 1, 12}, {k, 1, n}] // Flatten (* Amiram Eldar, May 09 2024 *)
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PROG
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(PARI) T(n, k) = sum(j=1, n, eulerphi(k*j))/eulerphi(k);
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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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