A071068 - OEIS

%I #28 Aug 13 2020 12:35:42

%S 0,1,1,2,1,2,2,3,2,2,2,4,3,3,3,5,4,4,3,6,4,5,4,7,5,5,5,7,5,5,5,8,6,7,

%T 6,11,7,7,7,11,8,8,9,13,10,8,8,13,10,8,7,14,10,10,7,13,10,11,9,15,11,

%U 11,11,15,11,11,11,18,12,13,11,21,13,14,13,20,14,13,14,20,16,13,13,22,15

%N Number of ways to write n as a sum of two unordered squarefree numbers.

%C The natural density of the squarefree numbers is 6/Pi^2, so An < a(n) < Bn for all large enough n with A < 6/Pi^2 - 1/2 and B > 3/Pi^2. The Schnirelmann density of the squarefree numbers is 53/88 > 1/2, and so a(n) > 0 for all n > 1 (in fact, a(n+1) >= 9n/88). It follows from Theoreme 3 bis. in Cohen, Dress, & El Marraki along with finite checking up to 16089908 that 0.10792n < a(n) < 0.303967n for n > 36. (The lower bound holds for n > 1.) - _Charles R Greathouse IV_, Feb 02 2016

%H T. D. Noe, <a href="/A071068/b071068.txt">Table of n, a(n) for n = 1..10000</a>

%H Henri Cohen, Francois Dress, and Mahomed El Marraki, <a href="https://projecteuclid.org/euclid.facm/1229618741">Explicit estimates for summatory functions linked to the Möbius μ-function</a>, Funct. Approx. Comment. Math. 37:1 (2007), pp. 51-63.

%F a(n) = sum(i=1..floor(n/2), abs(mu(i)*mu(n-i)) ). - _Wesley Ivan Hurt_, May 20 2013

%e 12=1+11=2+10=5+7=6+6 hence a(12)=4.

%t Table[Sum[Abs[MoebiusMu[i] MoebiusMu[n - i]],{i, 1, Floor[n/2]}],{n, 1, 85}] (* _Indranil Ghosh_, Mar 10 2017 *)

%t Table[Count[IntegerPartitions[n,{2}],_?(AllTrue[#,SquareFreeQ]&)],{n,90}] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Aug 13 2020 *)

%o (PARI) a(n)=sum(i=1,n\2,issquarefree(i)&&issquarefree(n-i)) \\ _Charles R Greathouse IV_, May 21 2013

%o (PARI) list(lim)=my(n=lim\1); concat(0, ceil(Vec((Polrev(vector(n, k, issquarefree(k-1))) + O('x^(n+1)))^2)/2)) \\ _Charles R Greathouse IV_, May 21 2013

%Y Cf. A005117, A098235.

%K easy,nonn

%O 1,4

%A _Benoit Cloitre_, May 26 2002