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A101035
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Dirichlet inverse of the gcd-sum function (A018804).
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4
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1, -3, -5, 1, -9, 15, -13, 1, 4, 27, -21, -5, -25, 39, 45, 1, -33, -12, -37, -9, 65, 63, -45, -5, 16, 75, 4, -13, -57, -135, -61, 1, 105, 99, 117, 4, -73, 111, 125, -9, -81, -195, -85, -21, -36, 135, -93, -5, 36, -48, 165, -25, -105, -12, 189, -13, 185, 171, -117, 45, -121, 183, -52, 1, 225, -315, -133, -33, 225, -351, -141, 4
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history;
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OFFSET
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1,2
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LINKS
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_Reinhard Zumkeller_, Table of n, a(n) for n = 1..10000
G. P. Michon, Multiplicative Functions.
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FORMULA
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Multiplicative function with a(p)=1-2p and a(p^e)=(p-1)^2 when e>1 [p prime].
Dirichlet g.f. zeta(s)/zeta^2(s-1). - R. J. Mathar, Apr 10 2011
a(n)=Sum{d|n} tau_{-2}(d)*d, where tau_{-2} is A007427. - Enrique Pérez Herrero, Jan 19 2013
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EXAMPLE
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a(4)=1, a(8)=1, a(16)=1, a(32)=1, etc. because of the multiplicative definition for powers of 2.
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MATHEMATICA
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DirichletInverse[f_][1] = 1/f[1]; DirichletInverse[f_][n_] := DirichletInverse[f][n] = -1/f[1]*Sum[ f[n/d]*DirichletInverse[f][d], {d, Most[ Divisors[n]]}]; GCDSum[n_] := Sum[ GCD[n, k], {k, 1, n}]; Table[ DirichletInverse[ GCDSum][n], {n, 1, 72}](* From Jean-François Alcover, Dec 12 2011 *)
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PROG
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(Haskell)
a101035 n = product $ zipWith f (a027748_row n) (a124010_row n) where
f p 1 = 1 - 2 * p
f p e = (p - 1) ^ 2
-- Reinhard Zumkeller, Jul 16 2012
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CROSSREFS
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Cf. A018804, A055615, A046692, A023900, A007427, A053822, A053825, A053826.
Cf. A008683.
Sequence in context: A214229 A214728 A112752 * A204029 A026253 A138259
Adjacent sequences: A101032 A101033 A101034 * A101036 A101037 A101038
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KEYWORD
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easy,nice,sign,mult
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AUTHOR
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Gerard P. Michon, Nov 27 2004
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STATUS
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approved
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