|
| |
|
|
A343516
|
|
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).
|
|
12
|
|
|
|
1, 1, 3, 1, 4, 5, 1, 5, 8, 8, 1, 6, 12, 15, 9, 1, 7, 17, 26, 19, 15, 1, 8, 23, 42, 39, 35, 13, 1, 9, 30, 64, 74, 76, 34, 20, 1, 10, 38, 93, 130, 153, 90, 56, 21, 1, 11, 47, 130, 214, 287, 216, 152, 63, 27, 1, 12, 57, 176, 334, 506, 468, 379, 191, 86, 21
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,3
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f. of column k: Sum_{j>=1} phi(j) * x^j/(1 - x^j)^(k+1).
T(n,k) = Sum_{d|n} phi(n/d) * binomial(d+k-1, k).
|
|
|
EXAMPLE
|
T(4,2) = gcd(1,1,4) + gcd(1,2,4) + gcd(2,2,4) + gcd(1,3,4) + gcd(2,3,4) + gcd(3,3,4) + gcd(1,4,4) + gcd(2,4,4) + gcd(3,4,4) + gcd(4,4,4) = 1 + 1 + 2 + 1 + 1 + 1 + 1 + 2 + 1 + 4 = 15.
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
3, 4, 5, 6, 7, 8, 9, ...
5, 8, 12, 17, 23, 30, 38, ...
8, 15, 26, 42, 64, 93, 130, ...
9, 19, 39, 74, 130, 214, 334, ...
15, 35, 76, 153, 287, 506, 846, ...
13, 34, 90, 216, 468, 930, 1722, ...
|
|
|
MATHEMATICA
|
T[n_, k_] := DivisorSum[n, EulerPhi[n/#] * Binomial[k + # - 1, k] &]; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Apr 18 2021 *)
|
|
|
PROG
|
(PARI) T(n, k) = sumdiv(n, d, eulerphi(n/d)*binomial(d+k-1, k));
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
