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A282634 Recursive 2-parameter sequence allowing the Ramanujan's sum calculation. 6

%I #88 Aug 05 2019 02:01:03

%S 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 -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 10,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,4,0,2,0,-2,0,-4,0

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

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

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

%H Seiichi Manyama, <a href="/A282634/b282634.txt">Rows n=1..140 of triangle, flattened</a>

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

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

%F From _Seiichi Manyama_, Mar 05 2018: (Start)

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

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

%e The few first rows follow: c_n(t)

%e t 0 1 2 3 4 5 6 | t 1 2 3 4 5 6 7

%e n |n

%e 1 1; |1 1;

%e 2 1, -1; |2 -1, 1;

%e 3 2, -1, -1; |3 -1, -1, 2;

%e 4 2, 0, -2, 0; |4 0, -2, 0, 2;

%e 5 4, -1, -1, -1, -1; |5 -1, -1, -1, -1, 4;

%e 6 2, 1, -1, -2, -1, 1; |6 1, -1, -2, -1, 1, 2;

%e 7 6, -1, -1, -1, -1, -1, -1; |7 -1, -1, -1, -1, -1, -1, 6;

%e ... | ...

%e [Edited by _Seiichi Manyama_, Mar 05 2018]

%t b[n_, m_] := b[n, m] = If[n > 1, b[n - 1, m] - b[n - 1, m - n + 1], 0]

%t b[1, m_] := b[1, m] = If[m == 0, 1, 0]

%t nt[n_, t_] := Round[(n - 1)/2 - t/n]

%t a[n_, t_] := Sum[b[n, k*n + t], {k, 0, nt[n, t]}]

%t Flatten[Table[Table[a[n, m], {m, 0, n - 1}], {n, 1, 20}]]

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

%K sign,tabl

%O 1,4

%A _Gevorg Hmayakyan_, Feb 20 2017

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Last modified March 29 08:13 EDT 2024. Contains 371265 sequences. (Running on oeis4.)