A077438 Numbers k such that Sum_{d|k} mu(d) mu(n/d)^2 = -1.

4, 9, 25, 49, 121, 169, 289, 361, 529, 841, 900, 961, 1369, 1681, 1764, 1849, 2209, 2809, 3481, 3721, 4356, 4489, 4900, 5041, 5329, 6084, 6241, 6889, 7921, 9409, 10201, 10404, 10609, 11025, 11449, 11881, 12100, 12769, 12996, 16129, 16900

Offset: 1

Author

Benoit Cloitre, Nov 30 2002

Keywords

• nonn

Comments

From Robert G. Wilson v, Dec 28 2016: (Start)

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

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

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..1000 (first 800 terms from G. C. Greubel)

Crossrefs

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

Programs

Mathematica

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

PARI

isok(n) = sumdiv(n, d, moebius(d)*moebius(n/d)^2) == -1; \\ Michel Marcus, Nov 08 2013

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

Formula

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

From Amiram Eldar, Jun 16 2020: (Start)

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

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