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A372938
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Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Sum_{1 <= x_1, x_2, ..., x_k <= n} gcd(x_1, x_2, ..., x_k, n)^k.
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3
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1, 1, 3, 1, 7, 5, 1, 15, 17, 8, 1, 31, 53, 40, 9, 1, 63, 161, 176, 49, 15, 1, 127, 485, 736, 249, 119, 13, 1, 255, 1457, 3008, 1249, 795, 97, 20, 1, 511, 4373, 12160, 6249, 4991, 685, 208, 21, 1, 1023, 13121, 48896, 31249, 30555, 4801, 1856, 225, 27
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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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a(n) = Sum_{d|n} mu(n/d) * d^k * tau(d), where mu is the Moebius function A008683.
T(n,k) for a given k is multiplicative with T(p^e, k) = (e - e/p^k + 1) * p^(k*e).
Dirichlet g.f. of T(n, k) for a given k: zeta(s-k)^2/zeta(s).
Sum_{m=1..n} T(m, k) ~ (n^(k+1)/((k+1)*zeta(k+1))) * (log(n) + 2*gamma - 1/(k+1) - zeta'(k+1)/zeta(k+1)), where gamma is Euler's constant (A001620). (End)
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
3, 7, 15, 31, 63, 127, 255, ...
5, 17, 53, 161, 485, 1457, 4373, ...
8, 40, 176, 736, 3008, 12160, 48896, ...
9, 49, 249, 1249, 6249, 31249, 156249, ...
15, 119, 795, 4991, 30555, 185039, 1115115, ...
13, 97, 685, 4801, 33613, 235297, 1647085, ...
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MATHEMATICA
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f[p_, e_, k_] := (e - e/p^k + 1)*p^(k*e); T[1, k_] := 1; T[n_, k_] := Times @@ (f[First[#], Last[#], k] & /@ FactorInteger[n]); Table[T[k, n - k + 1], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, May 25 2024 *)
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PROG
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(PARI) T(n, k) = sumdiv(n, d, moebius(n/d)*d^k*numdiv(d));
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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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