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A054533
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Triangular array giving Ramanujan sum T(n,k) = c_n(k) = Sum_{m=1..n, (m,n)=1} exp(2 Pi i m k / n) for n >= 1 and 1 <= k <= n.
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24
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1, -1, 1, -1, -1, 2, 0, -2, 0, 2, -1, -1, -1, -1, 4, 1, -1, -2, -1, 1, 2, -1, -1, -1, -1, -1, -1, 6, 0, 0, 0, -4, 0, 0, 0, 4, 0, 0, -3, 0, 0, -3, 0, 0, 6, 1, -1, 1, -1, -4, -1, 1, -1, 1, 4, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 10, 0, 2, 0, -2, 0, -4, 0, -2, 0, 2, 0, 4, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 12, 1
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OFFSET
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1,6
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COMMENTS
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Periodicity: c_n(k+n) = c_n(k). See the Apostol reference p. 161.
Multiplicativity: c_n(k)*c_m(k) = c_{n*m}(k), if gcd(n,m) = 1. For the proof see the Hardy reference, p. 138.
Dirichlet g.f. for fixed k: D(n,s) := Sum_{n>=1} c_n(k)/n^s = sigma_{1-s}(k)/zeta(s) = sigma_{s-1}(k)/(k^(s-1)*zeta(s)) for s > 1, with sigma_m(k) the sum of the m-th power of the divisors of k. See the Hardy reference, eqs. (9.6.1) and (9.6.2), pp. 139-140, or Hardy-Wright, Theorem 292, p. 250.
Sum_{n>=1} c_n(k)/n = 0. See the Hardy reference, p. 141. (End)
Fredman (1975) proved that the number S(n, k, v) of vectors (a_0, ..., a_{n-1}) of nonnegative integer components that satisfy a_0 + ... + a_{n-1} = k and Sum_{i=0..n-1} i*a_i = v (mod n) is given by S(n, k, v) = (1/(n + k)) * Sum_{d | gcd(n, k)} T(d, v) * binomial((n + k)/d, k/d) = S(k, n, v). This was also proved by Elashvili et al. (1999), who also proved that S(n, k, v) = Sum_{d | gcd(n, k, v)} S(n/d, k/d, 1). Here, S(n, k, 1) = A051168(n + k, k). - Petros Hadjicostas, Jul 09 2019
We have T(n, k) = c_n(k) = Sum_{m=1..n, (m,n)=1} exp(2 Pi i m k / n) and A054532(n, k) = c_k(n) = Sum_{m=1..k, (m,k)=1} exp(2 Pi i m n / k) for n >= 1 and 1 <= k <= n. - Petros Hadjicostas, Jul 27 2019
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REFERENCES
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T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, pp. 160-161.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, pp. 137-139.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. Fifth ed., Oxford Science Publications, Clarendon Press, Oxford, 2003, pp. 237-238.
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LINKS
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J. C. Kluyver, Some formulae concerning the integers less than n and prime to n, in: KNAW, Proceedings, 9 I, 1906, Amsterdam, 1906, pp. 408-414. [See bottom of p. 410, where the author proves that Sum cos(2*Pi*q*v/n) = mu(n/D) * phi(n) /phi(n/D), where D is the gcd of n and q. The summation is over integers v "less than n and prime to n" (top of p. 408).]
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FORMULA
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T(n, k) = Sum_{m=1..n, gcd(m,n) = 1} exp(2*Pi*i*m*k / n), n >= 1, 1 <= k <= n, where i is the imaginary unit.
T(n, k) = Sum_{d | gcd(n,k)} d*Moebius(n/d), n >= 1, 1 <= k <= n.
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EXAMPLE
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Triangle begins
1;
-1, 1;
-1, -1, 2;
0, -2, 0, 2;
-1, -1, -1, -1, 4;
1, -1, -2, -1, 1, 2;
-1, -1, -1, -1, -1, -1, 6;
0, 0, 0, -4, 0, 0, 0, 4;
0, 0, -3, 0, 0, -3, 0, 0, 6;
1, -1, 1, -1, -4, -1, 1, -1, 1, 4;
-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 10;
0, 2, 0, -2, 0, -4, 0, -2, 0, 2, 0, 4;
-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 12;
...
Periodicity and multiplicativity: c_6(k) = c_2(k)*c_3(k), e.g.: 2 = c_6(6) = c_2(6)*c_3(6) = c_2(2)*c_3(3) = 1*2 = 2. - Wolfdieter Lang, Jan 05 2017
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MATHEMATICA
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c[k_, n_] := Sum[ If[GCD[m, k] == 1, Exp[2 Pi*I*m*n/k], 0], {m, 1, k}]; A054533 = Flatten[ Table[c[n, k] // FullSimplify, {n, 1, 14}, {k, 1, n}] ] (* Jean-François Alcover, Jun 27 2012 *)
(* to get the triangle in the example above *)
FormTable[Table[c[n, k] // FullSimplify, {n, 1, 13}, {k, 1, n}]]
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PROG
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(PARI) T(n, k) = sumdiv(gcd(n, k), d, d*moebius(n/d));
tabl(nn) = {for(n=1, nn, for(k=1, n, print1(T(n, k), ", "); ); print(); ); }; \\ Michel Marcus, Jun 14 2018
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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