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%I #116 Feb 06 2024 12:04:52
%S 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,
%T 26,0,0,-29,-30,-31,0,33,34,35,0,-37,38,39,0,-41,-42,-43,0,0,46,-47,0,
%U 0,0,51,0,-53,0,55,0,57,58,-59,0,-61,62,0,0,65,-66,-67,0,69,-70,-71,0
%N a(n) = n * mu(n), where mu is the Möbius function A008683.
%C Dirichlet inverse of n (A000027).
%C Absolute values give n if n is squarefree, otherwise 0.
%C a(n) is multiplicative because both mu(n) and n are. - _Mitch Harris_, Jun 09 2005
%C a(n) is multiplicative with a(p^1) = -p, a(p^e) = 0 if e > 1. - _David W. Wilson_, Jun 12 2005
%C Negative of the Moebius number of the dihedral group of order 2n. - _Eric M. Schmidt_, Jul 28 2013
%H Antti Karttunen, <a href="/A055615/b055615.txt">Table of n, a(n) for n = 1..20000</a> (first 1000 terms from T. D. Noe)
%H Mats Granvik, <a href="http://math.stackexchange.com/questions/156035/primes-approximated-by-eigenvalues">Primes approximated by eigenvalues</a>.
%H Mats Granvik, <a href="http://pastebin.com/neU7vZme">Mobius function times n approximated by eigenvalues</a>.
%F a(n) = n * A008683(n).
%F Dirichlet g.f.: 1/zeta(s-1).
%F Multiplicative with a(p^e) = -p*0^(e-1), e>0 and p prime. - _Reinhard Zumkeller_, Jul 17 2003
%F 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
%F Dirichlet generating function for the absolute value: zeta(s-1)/zeta(2s-2). - _Franklin T. Adams-Watters_, Sep 11 2005
%F G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} k*A(x^k). - _Ilya Gutkovskiy_, May 11 2019
%F Sum_{k=1..n} abs(a(k)) ~ 3*n^2/Pi^2. - _Amiram Eldar_, Feb 02 2024
%e 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 + ...
%p with(numtheory): A055615:=n->n*mobius(n): seq(A055615(n), n=1..100); # _Wesley Ivan Hurt_, Nov 18 2014
%t Table[n MoebiusMu[n], {n,80}] (* _Harvey P. Dale_, May 26 2011 *)
%o (PARI) {a(n) = if( n<1, 0, n * moebius(n))};
%o (PARI) {a(n) = if( n<1, 0, direuler(p=2, n, 1 - p*X)[n]))};
%o (Magma) [n*MoebiusMu(n): n in [1..80]]; // _Vincenzo Librandi_, Nov 19 2014
%o (Haskell)
%o a055615 n = a008683 n * n -- _Reinhard Zumkeller_, Sep 04 2015
%o (SageMath) [n*moebius(n) for n in (1..100)] # _G. C. Greubel_, May 24 2022
%o (Python)
%o from sympy import mobius
%o def A055615(n): return n*mobius(n) # _Chai Wah Wu_, Apr 01 2023
%Y Moebius transform of A023900.
%Y Cf. A000027 (Dirichlet inverse), A061669 (sum with it).
%Y Cf. A062004.
%Y Cf. A013929 (positions of 0's), A068340 (partial sums), A261869 (first differences), A261890 (second differences).
%Y Cf. A008683, A334657, A334659, A334660.
%K sign,easy,nice,mult
%O 1,2
%A _Michael Somos_, Jun 04 2000
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