login
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).
5
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
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 A374839 A369892
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, May 18 2024
STATUS
approved