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).

1, 1, 3, 1, 7, 7, 1, 15, 25, 11, 1, 31, 79, 55, 21, 1, 63, 241, 239, 121, 21, 1, 127, 727, 991, 621, 175, 43, 1, 255, 2185, 4031, 3121, 1185, 337, 43, 1, 511, 6559, 16255, 15621, 7471, 2395, 439, 61, 1, 1023, 19681, 65279, 78121, 45801, 16801, 3823, 673, 63

Offset

1,3

Links

Table of n, a(n) for n=1..55.

Formula

T(n,k) = Sum_{d|n} mu(n/d) * (n/d) * sigma_{k+1}(d).

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.

From Amiram Eldar, May 25 2024: (Start)

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).

Dirichlet g.f. of T(n, k) for a given k: zeta(s)*zeta(s-k-1)/zeta(s-1).

Sum_{m=1..n} T(m, k) ~ c * n^(k+2) / (k+2), where c = zeta(k+2)/zeta(k+1). (End)

Example

Square array begins:
   1,   1,    1,     1,      1,      1, ...
   3,   7,   15,    31,     63,    127, ...
   7,  25,   79,   241,    727,   2185, ...
  11,  55,  239,   991,   4031,  16255, ...
  21, 121,  621,  3121,  15621,  78121, ...
  21, 175, 1185,  7471,  45801, 277495, ...

Mathematica

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 *)

Prog

(PARI) T(n, k) = sumdiv(n, d, moebius(n/d)*n/d*sigma(d, k+1));

Crossrefs

Columns k=1..5 give A057660, A350156, A372952, A372961, A371878.

Main diagonal gives A372969.

Sequence in context: A322753 A136035 A132307 * A188463 A369892 A359576

Adjacent sequences: A372965 A372966 A372967 * A372969 A372970 A372971

Keyword

nonn,tabl

Author

Seiichi Manyama, May 18 2024

Status

approved