%I #40 Jun 16 2020 05:35:19

%S 4,9,25,49,121,169,289,361,529,841,900,961,1369,1681,1764,1849,2209,

%T 2809,3481,3721,4356,4489,4900,5041,5329,6084,6241,6889,7921,9409,

%U 10201,10404,10609,11025,11449,11881,12100,12769,12996,16129,16900

%N Numbers k such that Sum_{d|k} mu(d) mu(n/d)^2 = -1.

%C From _Robert G. Wilson v_, Dec 28 2016: (Start)

%C Union of {A000040, A007304, A046387, A123321, A115343, etc}^2 = Union of {A001248, A162143, etc} = A030059(n)^2.

%C Number of terms < 10^k: 2, 4, 12, 30, 98, 303, 957, ..., . (End)

%H Robert G. Wilson v, <a href="/A077438/b077438.txt">Table of n, a(n) for n = 1..1000</a> (first 800 terms from G. C. Greubel)

%F a(n) = A030059(n)^2.

%F From _Amiram Eldar_, Jun 16 2020: (Start)

%F Sum_{k>=1} 1/a(k) = 9/(2*Pi^2) = A088245.

%F Sum_{k>=1} 1/a(k)^2 = 15/(2*Pi^4). (End)

%t fQ[n_] := Block[{d = Divisors@ n}, Plus @@ (MoebiusMu[#] MoebiusMu[n/#]^2 & /@ d) == -1]; Select[Range@17000, fQ] (* _Robert G. Wilson v_, Dec 28 2016 *)

%o (PARI) isok(n) = sumdiv(n, d, moebius(d)*moebius(n/d)^2) == -1; \\ _Michel Marcus_, Nov 08 2013

%o (PARI) is(n)=if(!issquare(n,&n), return(0)); my(f=factor(n)[,2]); #f%2 && vecmax(f)==1 \\ _Charles R Greathouse IV_, Oct 16 2015

%Y Cf. A000040, A001248, A007304, A030059, A046387, A088245, A115343, A123321, A162143.

%K nonn

%O 1,1

%A _Benoit Cloitre_, Nov 30 2002