A372968 - OEIS

%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