%I #45 Jun 06 2023 07:42:17

%S 1,2,0,3,2,0,0,4,1,4,0,0,2,0,0,5,2,2,0,6,0,0,0,0,3,4,0,0,2,0,0,6,0,4,

%T 0,3,2,0,0,8,2,0,0,0,2,0,0,0,1,6,0,6,2,0,0,0,0,4,0,0,2,0,0,7,4,0,0,6,

%U 0,0,0,4,2,4,0,0,0,0,0,10,1,4,0,0,4,0,0,0,2,4,0,0,0,0,0,0,2,2,0,9,2,0,0,8,0

%N a(n) = Sum_{d|n} Kronecker(-1, d).

%H G. C. Greubel, <a href="/A035184/b035184.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) is multiplicative with a(2^e) = e + 1, a(p^e) = e + 1 if p == 1 (mod 4), a(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4). - _Michael Somos_, Jan 05 2012

%F a(4*n + 3) = a(9*n + 3) = a(9*n + 6) = 0. a(4*n + 1) = A008441(n). a(8*n + 1) = A113407(n). a(8*n + 5) = 2 * A053692(n). a(12*n + 1) = A002175(n). a(12*n + 5) = 2 * A121444(n).

%F Dirichlet g.f.: zeta(s)*beta(s)/(1 - 2^(-s)), where beta is the Dirichlet beta function. - _Ralf Stephan_, Mar 27 2015

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/2 = 1.570796... (A019669). - _Amiram Eldar_, Oct 17 2022

%e G.f. = x + 2*x^2 + 3*x^4 + 2*x^5 + 4*x^8 + x^9 + 4*x^10 + 2*x^13 + 5*x^16 + 2*x^17 + ...

%t a[n_] := DivisorSum[n, KroneckerSymbol[-1, #] &]; Array[a, 105] (* _Jean-François Alcover_, Dec 02 2015 *)

%o (PARI) {a(n) = if( n<1, 0, direuler( p=2, n, 1/((1 - X) * (1 - kronecker( -1, p) * X))) [n])}; /* _Michael Somos_, Jan 05 2012 */

%o (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, kronecker( -1, d)))}; /* _Michael Somos_, Jan 05 2012 */

%Y Cf. A002175, A008441, A019669, A053692, A113407, A121444.

%Y Inverse Moebius transform of A034947.

%Y Sum_{d|n} Kronecker(k, d): A035143..A035181 (k=-47..-9, skipping numbers that are not cubefree), A035182 (k=-7), A192013 (k=-6), A035183 (k=-5), A002654 (k=-4), A002324 (k=-3), A002325 (k=-2), this sequence (k=-1), A000012 (k=0), A000005 (k=1), A035185 (k=2), A035186 (k=3), A001227 (k=4), A035187..A035229 (k=5..47, skipping numbers that are not cubefree).

%K nonn,easy,mult

%O 1,2

%A _N. J. A. Sloane_