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A372938 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.

%I A372938 #18 May 25 2024 12:24:54
%S A372938 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,
%T A372938 249,119,13,1,255,1457,3008,1249,795,97,20,1,511,4373,12160,6249,4991,
%U A372938 685,208,21,1,1023,13121,48896,31249,30555,4801,1856,225,27
%N A372938 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.
%F A372938 a(n) = Sum_{d|n} mu(n/d) * d^k * tau(d), where mu is the Moebius function A008683.
%F A372938 From _Amiram Eldar_, May 25 2024: (Start)
%F A372938 T(n,k) for a given k is multiplicative with T(p^e, k) = (e - e/p^k + 1) * p^(k*e).
%F A372938 Dirichlet g.f. of T(n, k) for a given k: zeta(s-k)^2/zeta(s).
%F A372938 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)
%e A372938 Square array begins:
%e A372938    1,   1,   1,    1,     1,      1,       1, ...
%e A372938    3,   7,  15,   31,    63,    127,     255, ...
%e A372938    5,  17,  53,  161,   485,   1457,    4373, ...
%e A372938    8,  40, 176,  736,  3008,  12160,   48896, ...
%e A372938    9,  49, 249, 1249,  6249,  31249,  156249, ...
%e A372938   15, 119, 795, 4991, 30555, 185039, 1115115, ...
%e A372938   13,  97, 685, 4801, 33613, 235297, 1647085, ...
%t A372938 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 *)
%o A372938 (PARI) T(n,k) = sumdiv(n, d, moebius(n/d)*d^k*numdiv(d));
%Y A372938 Columns k=1..4 give: A018804, A360428, A372928, A372931.
%Y A372938 Main diagonal gives A372939.
%Y A372938 Cf. A000005, A001620, A008683, A343510.
%K A372938 nonn,tabl
%O A372938 1,3
%A A372938 _Seiichi Manyama_, May 17 2024