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A344479 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). 8
1, 1, 3, 1, 5, 6, 1, 9, 12, 10, 1, 17, 30, 24, 15, 1, 33, 84, 76, 37, 21, 1, 65, 246, 276, 141, 61, 28, 1, 129, 732, 1060, 649, 267, 80, 36, 1, 257, 2190, 4164, 3165, 1417, 400, 112, 45, 1, 513, 6564, 16516, 15697, 8091, 2528, 624, 145, 55, 1, 1025, 19686, 65796, 78261, 47521, 17128, 4432, 885, 189, 66 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
FORMULA
G.f. of column k: (1/(1 - x)) * Sum_{i>=1} phi(i) * ( Sum_{j=1..k} A008292(k, j) * x^(i*j) )/(1 - x^i)^k.
T(n,k) = Sum_{j=1..n} phi(j) * floor(n/j)^k.
EXAMPLE
G.f. of column 3: (1/(1 - x)) * Sum_{i>=1} phi(i) * (x^i + 4*x^(2*i) + x^(3*i))/(1 - x^i)^3.
Square array begins:
1, 1, 1, 1, 1, 1, ...
3, 5, 9, 17, 33, 65, ...
6, 12, 30, 84, 246, 732, ...
10, 24, 76, 276, 1060, 4164, ...
15, 37, 141, 649, 3165, 15697, ...
21, 61, 267, 1417, 8091, 47521, ...
MATHEMATICA
T[n_, k_] := Sum[EulerPhi[j] * Quotient[n, j]^k, {j, 1, n}]; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, May 22 2021 *)
PROG
(PARI) T(n, k) = sum(j=1, n, eulerphi(j)*(n\j)^k);
CROSSREFS
Columns k=1..5 give A000217, A018806, A344522, A344523, A344524.
T(n,n) gives A344525.
Sequence in context: A113445 A108283 A208904 * A209754 A140950 A256504
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, May 22 2021
STATUS
approved

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Last modified June 17 12:36 EDT 2024. Contains 373445 sequences. (Running on oeis4.)