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

%I A282634 #88 Aug 05 2019 02:01:03
%S A282634 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,
%T A282634 -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,
%U A282634 10,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,4,0,2,0,-2,0,-4,0
%N A282634 Recursive 2-parameter sequence allowing the Ramanujan's sum calculation.
%C A282634 a(n,0) = phi(n), where phi(n) is Euler's totient function A000010(n).
%C A282634 a(n,1) = mu(n), where mu(n) is the Möbius function A008683(n).
%H A282634 Seiichi Manyama, <a href="/A282634/b282634.txt">Rows n=1..140 of triangle, flattened</a>
%H A282634 Gevorg Hmayakyan, <a href="https://oeis.org/w/images/9/93/Moebius_and_Totient.pdf">On The Moebius and Euler Totient Functions Calculation</a>.
%H A282634 Charles A. Nicol, <a href="http://www.pnas.org/content/39/9/963">On Restricted Partitions and a Generalization Of The Euler Totient and The Moebius Function</a>, PNAS 39(9) (1953), 963-968.
%F A282634 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].
%F A282634 a(n,t) = c_n(t) for t >= 1, where c_n(t) is a Ramanujan's sum A054533.
%F A282634 a(n,t) = a(n,-t)
%F A282634 From _Seiichi Manyama_, Mar 05 2018: (Start)
%F A282634 a(n,t) = c_n(n-t) = Sum_{d | gcd(n,n-t)} d*mu(n/d) for 0 <= t <= n-1.
%F A282634 So a(n,t) = Sum_{d | gcd(n,t)} d*mu(n/d) for 1 <= t <= n-1. (End)
%e A282634 The few first rows follow:            c_n(t)
%e A282634   t   0   1   2   3   4   5   6     |  t   1   2   3   4   5   6   7
%e A282634 n                                   |n
%e A282634 1     1;                            |1     1;
%e A282634 2     1, -1;                        |2    -1,  1;
%e A282634 3     2, -1, -1;                    |3    -1, -1,  2;
%e A282634 4     2,  0, -2,  0;                |4     0, -2,  0,  2;
%e A282634 5     4, -1, -1, -1, -1;            |5    -1, -1, -1, -1,  4;
%e A282634 6     2,  1, -1, -2, -1,  1;        |6     1, -1, -2, -1,  1,  2;
%e A282634 7     6, -1, -1, -1, -1, -1, -1;    |7    -1, -1, -1, -1, -1, -1,  6;
%e A282634       ...                           |     ...
%e A282634 [Edited by _Seiichi Manyama_, Mar 05 2018]
%t A282634 b[n_, m_] := b[n, m] = If[n > 1, b[n - 1, m] - b[n - 1, m - n + 1], 0]
%t A282634 b[1, m_] := b[1, m] = If[m == 0, 1, 0]
%t A282634 nt[n_, t_] := Round[(n - 1)/2 - t/n]
%t A282634 a[n_, t_] := Sum[b[n, k*n + t], {k, 0, nt[n, t]}]
%t A282634 Flatten[Table[Table[a[n, m], {m, 0, n - 1}], {n, 1, 20}]]
%Y A282634 Cf. A000010 (phi(n)), A008683 (mu(n)), A054532, A054533, A054534, A054535, A231599.
%K A282634 sign,tabl
%O A282634 1,4
%A A282634 _Gevorg Hmayakyan_, Feb 20 2017