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A282634 Recursive 2-parameter sequence allowing the Ramanujan's sum calculation. 6
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

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).

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.

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)

EXAMPLE

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]

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}]]

CROSSREFS

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

Sequence in context: A035192 A229653 A089062 * A039980 A306660 A194529

Adjacent sequences:  A282631 A282632 A282633 * A282635 A282636 A282637

KEYWORD

sign,tabl

AUTHOR

Gevorg Hmayakyan, Feb 20 2017

STATUS

approved

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Last modified October 22 10:24 EDT 2019. Contains 328317 sequences. (Running on oeis4.)