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

1,2

Links

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

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

Example

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

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 *)

Prog

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

Crossrefs

Cf. A070548, A070549, A081239, A081240.

Sequence in context: A224487 A185342 A276985 * A333823 A207254 A207403

Adjacent sequences: A081235 A081236 A081237 * A081239 A081240 A081241

Keyword

nonn

Author

Reinhard Zumkeller, Mar 11 2003

Status

approved