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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 A373572 A306660 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 September 14 09:44 EDT 2024. Contains 375921 sequences. (Running on oeis4.)