A081240 -id:A081240 - cp's OEIS frontend

A081239 #{(i,j): mu(i)*mu(j) = 0, 1<=i,j<=n}, where mu=A008683 (Moebius function).

0, 0, 0, 7, 9, 11, 13, 28, 45, 51, 57, 80, 88, 96, 104, 135, 145, 180, 192, 231, 245, 259, 273, 320, 369, 387, 440, 495, 517, 539, 561, 624, 648, 672, 696, 767, 793, 819, 845, 924, 952, 980, 1008, 1095, 1184, 1216, 1248, 1343, 1440, 1539, 1577, 1680, 1720

Offset: 1

Reinhard Zumkeller, Mar 11 2003

nonn

A081238(n) + a(n) + A081240(n) = n^2;

a(n) = a(n-1) + 2*n + 1 iff mu(n) = 0.

			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)=28, as there are
6 .. +1 ..... | + - - 0 - + - 0 28 '0's in the 8x8-square
7 .. -1 ..... | - + + 0 + - + 0
8 ... 0 ..... | 0 0 0 0 0 0 0 0.

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

Cf. A057627.

a081239 n = length [() | u <- [1..n], v <- [1..n],
                         a008683 u * a008683 v == 0]
-- Reinhard Zumkeller, Aug 03 2012

a(n) = n^2 - A013928(n+1)^2. - Vladeta Jovovic, Mar 12 2003

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

Original entry on oeis.org

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

Reinhard Zumkeller, Mar 11 2003

nonn

			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

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

Cf. A070548, A070549, A081239, A081240.

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

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

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 *)

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

cp's OEIS Frontend

A081239 #{(i,j): mu(i)*mu(j) = 0, 1<=i,j<=n}, where mu=A008683 (Moebius function).

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