A327029 T(n, k) = Sum_{d|n} phi(d) * A008284(n/d, k) for n >= 1, T(0, 0) = 1. Triangle read by rows for 0 <= k <= n.

1, 0, 1, 0, 2, 1, 0, 3, 1, 1, 0, 4, 3, 1, 1, 0, 5, 2, 2, 1, 1, 0, 6, 6, 4, 2, 1, 1, 0, 7, 3, 4, 3, 2, 1, 1, 0, 8, 8, 6, 6, 3, 2, 1, 1, 0, 9, 6, 9, 6, 5, 3, 2, 1, 1, 0, 10, 11, 10, 10, 8, 5, 3, 2, 1, 1, 0, 11, 5, 10, 11, 10, 7, 5, 3, 2, 1, 1

Offset

0,5

Comments

Dirichlet convolution of phi(n) and A008284(n,k) for n >= 1. - Richard L. Ollerton, May 07 2021

Links

Table of n, a(n) for n=0..77.

Formula

From Richard L. Ollerton, May 07 2021: (Start)

For n >= 1, T(n,k) = Sum_{i=1..n} A008284(gcd(n,i),k).

For n >= 1, T(n,k) = Sum_{i=1..n} A008284(n/gcd(n,i),k)*phi(gcd(n,i))/phi(n/gcd(n,i)). (End)

Example

Triangle starts:
[0] [1]
[1] [0, 1]
[2] [0, 2, 1]
[3] [0, 3, 1, 1]
[4] [0, 4, 3, 1, 1]
[5] [0, 5, 2, 2, 1, 1]
[6] [0, 6, 6, 4, 2, 1, 1]
[7] [0, 7, 3, 4, 3, 2, 1, 1]
[8] [0, 8, 8, 6, 6, 3, 2, 1, 1]
[9] [0, 9, 6, 9, 6, 5, 3, 2, 1, 1]

Prog

(SageMath)
def DivisorTriangle(f, T, Len, w = None):
    D = [[1]]
    for n in (1..Len-1):
        r = lambda k: [f(d)*T(n//d, k) for d in divisors(n)]
        L = [sum(r(k)) for k in (0..n)]
        if w != None: L = [*map(lambda v: v * w(n), L)]
        D.append(L)
    return D
DivisorTriangle(euler_phi, A008284, 10)

Crossrefs

Cf. A008284, A000010, A078392 (row sums), A282750.

Sequence in context: A285037 A264422 A176808 * A167192 A335435 A363048

Adjacent sequences: A327026 A327027 A327028 * A327030 A327031 A327032

Keyword

nonn,tabl

Author

Peter Luschny, Aug 24 2019

Status

approved