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A055615
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a(n) = n * mu(n), where mu is the Möbius function A008683.
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78
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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.
a(n) is multiplicative because both mu(n) and n are. - Mitch Harris, Jun 09 2005
a(n) is multiplicative with a(p^1) = -p, a(p^e) = 0 if e > 1. - David W. Wilson, Jun 12 2005
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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Antti Karttunen, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)
Mats Granvik, Primes approximated by eigenvalues
Mats Granvik, Mobius function times n approximated by eigenvalues
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FORMULA
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a(n) = n * A008683(n).
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
(SageMath) [n*moebius(n) for n in (1..100)] # G. C. Greubel, May 24 2022
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CROSSREFS
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Moebius transform of A023900.
Cf. A000027 (Dirichlet inverse), A061669 (sum with it).
Cf. A008683, A062004.
Cf. A013929 (positions of 0's), A068340 (partial sums), A261869 (first differences), A261890 (second differences).
Sequence in context: A248092 A145105 A140700 * A243059 A332845 A351079
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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