A081240 a(n) = #{(i,j): mu(i)*mu(j) = 1, 1<=i,j<=n}, where mu = A008683 (Moebius function).

1, 2, 5, 5, 10, 13, 20, 20, 20, 25, 34, 34, 45, 52, 61, 61, 74, 74, 89, 89, 100, 113, 130, 130, 130, 145, 145, 145, 164, 185, 208, 208, 225, 244, 265, 265, 290, 313, 338, 338, 365, 394, 425, 425, 425, 452, 485, 485, 485, 485, 514, 514, 549, 549, 580, 580, 613

Offset: 1

Author

Reinhard Zumkeller, Mar 11 2003

Keywords

• nonn

Comments

A081238(n) + A081239(n) + a(n) = n^2;

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

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)=20, as there are
6 .. +1 ..... | + - - 0 - + - 0 20 '+1's in the 8x8-square
7 .. -1 ..... | - + + 0 + - + 0 (represented as '+')
8 ... 0 ..... | 0 0 0 0 0 0 0 0.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 520 terms from Reinhard Zumkeller)

Crossrefs

Cf. A008683, A070548, A070549.

Programs

Haskell

a081240 n = length [() | u <- [1..n], v <- [1..n],
                         a008683 u * a008683 v == 1]
-- Reinhard Zumkeller, Aug 03 2012

Mathematica

Table[Abs[Sum[Sqrt[MoebiusMu[i]],{i,1,n}]]^2,{n,60}] (* Enrique Pérez Herrero, Jul 30 2012 *)

Formula

a(n) = |Sum_{i=1..n} sqrt(mu(i))|^2. - Enrique Pérez Herrero, Jul 30 2012

a(n) = A070548(n)^2 + A070549(n)^2. - Jason Yuen, Aug 23 2024