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A055615
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a(n) = n*moebius(n) (cf. A008683).
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40
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1, -2, -3, 0, -5, 6, -7, 0, 0, 10, -11, 0, -13, 14, 15, 0, -17, 0, -19, 0, 21, 22, -23, 0, 0, 26, 0, 0, -29, -30, -31, 0, 33, 34, 35, 0, -37, 38, 39, 0, -41, -42, -43, 0, 0, 46, -47, 0, 0, 0, 51, 0, -53, 0, 55, 0, 57, 58, -59, 0, -61, 62, 0, 0, 65, -66, -67, 0, 69, -70, -71, 0
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OFFSET
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1,2
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COMMENTS
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Dirichlet inverse of n.
Absolute values give n if n is squarefree, otherwise 0.
Equals row sums of triangle A127507, A127475. - Gary W. Adamson, May 01 2010
Negative of the Moebius number of the dihedral group of order 2n. - Eric M. Schmidt, Jul 28 2013
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
Mats Granvik, Primes approximated by eigenvalues
Mats Granvik, Mobius function times n approximated by eigenvalues
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FORMULA
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Dirichlet g.f.: 1/zeta(s-1).
Multiplicative with a(p^e) = -p*0^(e-1), e>0 and p prime. - Reinhard Zumkeller, Jul 17 2003
Conjectures: lim b->1+ Sum n=1..inf a(n)*b^(-n) = -12 and lim b->1- Sum n=1..inf a(n)*b^n = -12 (+ indicates that b decreases to 1, - indicates it increases to 1), both considering that zeta(-1) = -1/12 and calculations (more generally mu(n)*n^s is Abel summable to zeta(-s)). - Gerald McGarvey, Sep 26 2004
Dirichlet generating function for the absolute value: zeta(s-1)/zeta(2s-2). - Franklin T. Adams-Watters, Sep 11 2005
G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} k*A(x^k). - Ilya Gutkovskiy, May 11 2019
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EXAMPLE
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G.f. = x - 2*x^2 - 3*x^3 - 5*x^5 + 6*x^6 - 7*x^7 + 10*x^10 - 11*x^11 - 13*x^13 + ...
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MAPLE
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with(numtheory): A055615:=n->n*mobius(n): seq(A055615(n), n=1..100); # Wesley Ivan Hurt, Nov 18 2014
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MATHEMATICA
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Table[n MoebiusMu[n], {n, 80}] (* Harvey P. Dale, May 26 2011 *)
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PROG
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(PARI) {a(n) = if( n<1, 0, n * moebius(n))};
(PARI) {a(n) = if( n<1, 0, direuler(p=2, n, 1 - p*X)[n]))};
(MAGMA) [n*MoebiusMu(n): n in [1..80]]; // Vincenzo Librandi, Nov 19 2014
(Haskell)
a055615 n = a008683 n * n -- Reinhard Zumkeller, Sep 04 2015
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CROSSREFS
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Cf. A000027. Moebius transform of A023900.
Cf. A008683, A062004.
Cf. A127475, A127507.
Cf. A068340 (partial sums), A261869 (first differences), A261890 (second differences).
Sequence in context: A248092 A145105 A140700 * A243059 A190621 A325314
Adjacent sequences: A055612 A055613 A055614 * A055616 A055617 A055618
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KEYWORD
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sign,easy,nice,mult
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AUTHOR
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Michael Somos, Jun 04 2000
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STATUS
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approved
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