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).
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
Offset: 1
Examples
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, ...
Links
- Seiichi Manyama, Antidiagonals n = 1..140, flattened
Crossrefs
Programs
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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 *) -
PARI
T(n, k) = sumdiv(n, d, eulerphi(n/d)*binomial(d+k-1, k));
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).