A282634 Recursive 2-parameter sequence allowing the Ramanujan's sum calculation.

1, 1, -1, 2, -1, -1, 2, 0, -2, 0, 4, -1, -1, -1, -1, 2, 1, -1, -2, -1, 1, 6, -1, -1, -1, -1, -1, -1, 4, 0, 0, 0, -4, 0, 0, 0, 6, 0, 0, -3, 0, 0, -3, 0, 0, 4, 1, -1, 1, -1, -4, -1, 1, -1, 1, 10, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 4, 0, 2, 0, -2, 0, -4, 0

Offset: 1

Author

Gevorg Hmayakyan, Feb 20 2017

Keywords

• sign
• tabl

Comments

a(n,0) = phi(n), where phi(n) is Euler's totient function A000010(n).

a(n,1) = mu(n), where mu(n) is the Möbius function A008683(n).

Examples

			The few first rows follow:            c_n(t)
  t   0   1   2   3   4   5   6     |  t   1   2   3   4   5   6   7
n                                   |n
1     1;                            |1     1;
2     1, -1;                        |2    -1,  1;
3     2, -1, -1;                    |3    -1, -1,  2;
4     2,  0, -2,  0;                |4     0, -2,  0,  2;
5     4, -1, -1, -1, -1;            |5    -1, -1, -1, -1,  4;
6     2,  1, -1, -2, -1,  1;        |6     1, -1, -2, -1,  1,  2;
7     6, -1, -1, -1, -1, -1, -1;    |7    -1, -1, -1, -1, -1, -1,  6;
      ...                           |     ...
[Edited by _Seiichi Manyama_, Mar 05 2018]
		

Links

• Seiichi Manyama, Rows n=1..140 of triangle, flattened
• Gevorg Hmayakyan, On The Moebius and Euler Totient Functions Calculation.
• Charles A. Nicol, On Restricted Partitions and a Generalization Of The Euler Totient and The Moebius Function, PNAS 39(9) (1953), 963-968.

Crossrefs

Cf. A000010 (phi(n)), A008683 (mu(n)), A054532, A054533, A054534, A054535, A231599.

Programs

Mathematica

b[n_, m_] := b[n, m] = If[n > 1, b[n - 1, m] - b[n - 1, m - n + 1], 0]
b[1, m_] := b[1, m] = If[m == 0, 1, 0]
nt[n_, t_] := Round[(n - 1)/2 - t/n]
a[n_, t_] := Sum[b[n, k*n + t], {k, 0, nt[n, t]}]
Flatten[Table[Table[a[n, m], {m, 0, n - 1}], {n, 1, 20}]]

Formula

a(n,t) = Sum(b(n, k*n + t), k=0..N(n, t)), where b(n,k) = A231599(n-1,k) and N(n,t) = [(n - 1)/2 - t/n].

a(n,t) = c_n(t) for t >= 1, where c_n(t) is a Ramanujan's sum A054533.

a(n,t) = a(n,-t)

From Seiichi Manyama, Mar 05 2018: (Start)

a(n,t) = c_n(n-t) = Sum_{d | gcd(n,n-t)} d*mu(n/d) for 0 <= t <= n-1.

So a(n,t) = Sum_{d | gcd(n,t)} d*mu(n/d) for 1 <= t <= n-1. (End)