OFFSET
1,4
COMMENTS
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
LINKS
T. D. Noe, Table of n, a(n) for n = 1..10000
Henri Cohen, Francois Dress, and Mahomed El Marraki, Explicit estimates for summatory functions linked to the Möbius μ-function, Funct. Approx. Comment. Math. 37:1 (2007), pp. 51-63.
FORMULA
a(n) = sum(i=1..floor(n/2), abs(mu(i)*mu(n-i)) ). - Wesley Ivan Hurt, May 20 2013
EXAMPLE
12=1+11=2+10=5+7=6+6 hence a(12)=4.
MATHEMATICA
Table[Sum[Abs[MoebiusMu[i] MoebiusMu[n - i]], {i, 1, Floor[n/2]}], {n, 1, 85}] (* Indranil Ghosh, Mar 10 2017 *)
Table[Count[IntegerPartitions[n, {2}], _?(AllTrue[#, SquareFreeQ]&)], {n, 90}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 13 2020 *)
PROG
(PARI) a(n)=sum(i=1, n\2, issquarefree(i)&&issquarefree(n-i)) \\ Charles R Greathouse IV, May 21 2013
(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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Benoit Cloitre, May 26 2002
STATUS
approved