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A127415
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Sum {1<=k<=n, gcd(k,n)=1}, A000217(k).
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2
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1, 1, 4, 7, 20, 16, 56, 50, 93, 80, 220, 110, 364, 224, 340, 372, 816, 354, 1140, 580, 966, 880, 2024, 820, 2200, 1456, 2304, 1666, 4060, 1240, 4960, 2856, 3850
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| From Wolfdieter Lang, Jun 14 2011: (Start)
Such sums are over a reduced residue system modulo n. See the Apostol reference, p. 133, for the definition or the wikipedia link given under A189918.
This sum over triangular numbers can be found using the results given in exercise 16 of the Apostol reference on p. 48, together with the definition of phi_1(n) and phi_2(n) from the exercise 15.
The result for n>=2 coincides with the formula given below, using for product(1-p,p|n) = mu(sqfk(n))*sfk(n)*phi(n)/n, with the definitions given there.
(End)
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REFERENCES
| T. Apostol, Introducton to Analytic Number Theory, Springer, 1986.
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FORMULA
| M * V where M = A054521 is an infinite lower triangular matrix and V = A000217: (1, 3, 6, 10,...).
From Wolfdieter Lang, May 17 2011: (Start)
a(n) = (n/(3!*2))*((2*n+3)*n + mu(sqfk(n))*sfk(n))*(phi(n)/n), n>=2, with sqfk(n) the squarefree kernel of n (the largest squarefree number dividing n, see A007947), the Moebius function mu(n)=A008683(n), and the Euler totient function phi(n)= A000010(n).
Note that phi(n)/n = A076512(n)/A109395(n) = phi(sqfk(n))/sqfk(n).
Proof via inclusion-exclusion.
(End)
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EXAMPLE
| a(6) = 16 since the relative primes of 6 are 1 and 5 and (1 + 15) = 16.
a(6) = (6/(3!*2))*(15*6 + 1*6)*(1/2)*(2/3)= 16.
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MATHEMATICA
| sfk[n_] := Times @@ (FactorInteger[n][[ All, 1]]); a[n_] := (n/(3!*2))*((2*n+3)*n + MoebiusMu[ sfk[n]]*sfk[n])*(EulerPhi[n] / n); a[1] = 1; Table[ a[n], {n, 1, 33}] (* From Jean-François Alcover, Oct 03 2011 *)
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PROG
| (PARI) a(n)=if(n<3, return(1)); my(s=factor(n)[, 1]); s=prod(i=1, #s, s[i]); (n/12)*((2*n+3)*n + moebius(s)*s)*(eulerphi(n)/n) \\ Charles R Greathouse IV, May 17, 2011
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CROSSREFS
| Cf. A000217, A023896, A076512/A109395, A189918.
Sequence in context: A182646 A190646 A133264 * A045548 A090879 A084404
Adjacent sequences: A127412 A127413 A127414 * A127416 A127417 A127418
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KEYWORD
| nonn,easy
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AUTHOR
| Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 13 2007
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EXTENSIONS
| More terms and formula by Wolfdieter Lang, May 17 2011.
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