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