A081238 #{(i,j): mu(i)*mu(j) = -1, 1 <= i <= n, 1 <= j <= n}, where mu=A008683 (Moebius function).

0, 2, 4, 4, 6, 12, 16, 16, 16, 24, 30, 30, 36, 48, 60, 60, 70, 70, 80, 80, 96, 112, 126, 126, 126, 144, 144, 144, 160, 176, 192, 192, 216, 240, 264, 264, 286, 312, 338, 338, 364, 390, 416, 416, 416, 448, 476, 476, 476, 476, 510, 510, 540, 540, 576, 576, 612, 648

Offset: 1

Author

Reinhard Zumkeller, Mar 11 2003

Keywords

• nonn

Examples

			n  mu(n)  n: 1 2 3 4 5 6 7 8
-  -----   +----------------->
1   +1     | + - - 0 - + - 0
2   -1     | - + + 0 + - + 0
3   -1     | - + + 0 + - + 0
4    0     | 0 0 0 0 0 0 0 0
5   -1     | - + + 0 + - + 0  a(8)=16, as there are
6   +1     | + - - 0 - + - 0  16 '-1's in the 8 X 8 square
7   -1     | - + + 0 + - + 0  (represented as '-')
8    0     | 0 0 0 0 0 0 0 0
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..500

Crossrefs

Cf. A070548, A070549, A081239, A081240.

Programs

Haskell

a081238 n = length [() | u <- [1..n], v <- [1..n],
                         a008683 u * a008683 v == -1]
-- Reinhard Zumkeller, Aug 03 2012

Maple

Nplus:= 0:
Nminus:=0:
for n from 1 to 100 do
  v:= numtheory:-mobius(n);
  if v = 1 then Nplus:= Nplus+1
  elif v = -1 then Nminus:= Nminus+1
  fi;
  A[n]:= 2*Nplus*Nminus;
od:
seq(A[n],n=1..100); # Robert Israel, Jan 08 2018

Mathematica

Nplus = Nminus = 0;
For[n = 1, n <= 100, n++, v = MoebiusMu[n];
     If[v == 1, Nplus++,
     If[v == -1, Nminus++]];
     a[n] = 2 Nplus Nminus];
Array[a, 100] (* Jean-François Alcover, Dec 16 2021, after Robert Israel *)

Formula

a(n) + A081239(n) + A081240(n) = n^2;

a(n) = a(n-1) iff mu(n) = 0.

a(n) = 2*A070548(n)*A070549(n). - Robert Israel, Jan 08 2018