

A071068


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


23



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



FORMULA

a(n) = sum(i=1..floor(n/2), abs(mu(i)*mu(ni)) ).  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) list(lim)=my(n=lim\1); concat(0, ceil(Vec((Polrev(vector(n, k, issquarefree(k1))) + O('x^(n+1)))^2)/2)) \\ Charles R Greathouse IV, May 21 2013


CROSSREFS



KEYWORD

easy,nonn


AUTHOR



STATUS

approved



