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%I #19 Dec 16 2021 04:13:41
%S 0,2,4,4,6,12,16,16,16,24,30,30,36,48,60,60,70,70,80,80,96,112,126,
%T 126,126,144,144,144,160,176,192,192,216,240,264,264,286,312,338,338,
%U 364,390,416,416,416,448,476,476,476,476,510,510,540,540,576,576,612,648
%N #{(i,j): mu(i)*mu(j) = -1, 1 <= i <= n, 1 <= j <= n}, where mu=A008683 (Moebius function).
%H Reinhard Zumkeller, <a href="/A081238/b081238.txt">Table of n, a(n) for n = 1..500</a>
%F a(n) + A081239(n) + A081240(n) = n^2;
%F a(n) = a(n-1) iff mu(n) = 0.
%F a(n) = 2*A070548(n)*A070549(n). - _Robert Israel_, Jan 08 2018
%e n mu(n) n: 1 2 3 4 5 6 7 8
%e - ----- +----------------->
%e 1 +1 | + - - 0 - + - 0
%e 2 -1 | - + + 0 + - + 0
%e 3 -1 | - + + 0 + - + 0
%e 4 0 | 0 0 0 0 0 0 0 0
%e 5 -1 | - + + 0 + - + 0 a(8)=16, as there are
%e 6 +1 | + - - 0 - + - 0 16 '-1's in the 8 X 8 square
%e 7 -1 | - + + 0 + - + 0 (represented as '-')
%e 8 0 | 0 0 0 0 0 0 0 0
%p Nplus:= 0:
%p Nminus:=0:
%p for n from 1 to 100 do
%p v:= numtheory:-mobius(n);
%p if v = 1 then Nplus:= Nplus+1
%p elif v = -1 then Nminus:= Nminus+1
%p fi;
%p A[n]:= 2*Nplus*Nminus;
%p od:
%p seq(A[n],n=1..100); # _Robert Israel_, Jan 08 2018
%t Nplus = Nminus = 0;
%t For[n = 1, n <= 100, n++, v = MoebiusMu[n];
%t If[v == 1, Nplus++,
%t If[v == -1, Nminus++]];
%t a[n] = 2 Nplus Nminus];
%t Array[a, 100] (* _Jean-François Alcover_, Dec 16 2021, after _Robert Israel_ *)
%o (Haskell)
%o a081238 n = length [() | u <- [1..n], v <- [1..n],
%o a008683 u * a008683 v == -1]
%o -- _Reinhard Zumkeller_, Aug 03 2012
%Y Cf. A070548, A070549, A081239, A081240.
%K nonn
%O 1,2
%A _Reinhard Zumkeller_, Mar 11 2003
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