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.

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

Offset: 1

Author

Seiichi Manyama, May 17 2024

Keywords

• nonn
• tabl

Examples

			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, ...
		

Crossrefs

Columns k=1..4 give: A018804, A360428, A372928, A372931.

Main diagonal gives A372939.

Cf. A000005, A001620, A008683, A343510.

Programs

Mathematica

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 *)

PARI

T(n,k) = sumdiv(n, d, moebius(n/d)*d^k*numdiv(d));

Formula

a(n) = Sum_{d|n} mu(n/d) * d^k * tau(d), where mu is the Moebius function A008683.

From Amiram Eldar, May 25 2024: (Start)

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)