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A199084
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a(n) = Sum_{k=1..n} (-1)^(k+1) gcd(k,n).
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8
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1, -1, 3, -4, 5, -5, 7, -12, 9, -9, 11, -20, 13, -13, 15, -32, 17, -21, 19, -36, 21, -21, 23, -60, 25, -25, 27, -52, 29, -45, 31, -80, 33, -33, 35, -84, 37, -37, 39, -108, 41, -65, 43, -84, 45, -45, 47, -160, 49, -65, 51, -100, 53, -81, 55, -156, 57
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OFFSET
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1,3
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COMMENTS
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The alternating sum analog of A018804.
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LINKS
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FORMULA
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Sum_{k=1..n} a(k) ~ (n^2/Pi^2) * (-log(n) - 2*gamma + 1/2 + 4*log(2)/3 + Pi^2/4 + zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620). - Amiram Eldar, Mar 30 2024
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MAPLE
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add((-1)^(k-1)* igcd(k, n), k=1..n) ;
end proc:
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MATHEMATICA
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altGCDSum[n_] := Sum[(-1)^(i + 1)GCD[i, n], {i, n}]; Table[altGCDSum[n], {n, 50}] (* Alonso del Arte, Nov 02 2011 *)
Total/@Table[(-1)^(k+1) GCD[k, n], {n, 60}, {k, n}] (* Harvey P. Dale, May 29 2013 *)
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PROG
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(PARI) a(n) = sum(k=1, n, (-1)^(k+1)*gcd(k, n)); \\ Michel Marcus, Jun 28 2023
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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