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A372968 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} n/gcd(x_1, x_2, ..., x_k, n).

%I A372968 #29 May 25 2024 12:29:26
%S A372968 1,1,3,1,7,7,1,15,25,11,1,31,79,55,21,1,63,241,239,121,21,1,127,727,
%T A372968 991,621,175,43,1,255,2185,4031,3121,1185,337,43,1,511,6559,16255,
%U A372968 15621,7471,2395,439,61,1,1023,19681,65279,78121,45801,16801,3823,673,63
%N A372968 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} n/gcd(x_1, x_2, ..., x_k, n).
%F A372968 T(n,k) = Sum_{d|n} mu(n/d) * (n/d) * sigma_{k+1}(d).
%F A372968 T(n,k) = Sum_{1 <= x_1, x_2, ..., x_k <= n} ( gcd(x_1, x_2, ..., x_{k-1}, n)/gcd(x_1, x_2, ..., x_k, n) )^k.
%F A372968 From _Amiram Eldar_, May 25 2024: (Start)
%F A372968 T(n,k) for a given k is multiplicative with T(p^e, k) = (p^((k+1)*(e+1)) - p^((k+1)*e+1) + p - 1)/(p^(k+1)-1).
%F A372968 Dirichlet g.f. of T(n, k) for a given k: zeta(s)*zeta(s-k-1)/zeta(s-1).
%F A372968 Sum_{m=1..n} T(m, k) ~ c * n^(k+2) / (k+2), where c = zeta(k+2)/zeta(k+1). (End)
%e A372968 Square array begins:
%e A372968    1,   1,    1,     1,      1,      1, ...
%e A372968    3,   7,   15,    31,     63,    127, ...
%e A372968    7,  25,   79,   241,    727,   2185, ...
%e A372968   11,  55,  239,   991,   4031,  16255, ...
%e A372968   21, 121,  621,  3121,  15621,  78121, ...
%e A372968   21, 175, 1185,  7471,  45801, 277495, ...
%t A372968 f[p_, e_, k_] := (p^((k + 1)*e + k + 1) - p^((k + 1)*e + 1) + p - 1)/(p^(k + 1) - 1); 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 A372968 (PARI) T(n, k) = sumdiv(n, d, moebius(n/d)*n/d*sigma(d, k+1));
%Y A372968 Columns k=1..5 give A057660, A350156, A372952, A372961, A371878.
%Y A372968 Main diagonal gives A372969.
%K A372968 nonn,tabl
%O A372968 1,3
%A A372968 _Seiichi Manyama_, May 18 2024