

A071068


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


22



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

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 μfunction, Funct. Approx. Comment. Math. 37:1 (2007), pp. 5163.


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) a(n)=sum(i=1, n\2, issquarefree(i)&&issquarefree(ni)) \\ Charles R Greathouse IV, May 21 2013
(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

Cf. A005117, A098235.
Sequence in context: A033265 A096004 A193495 * A240872 A328806 A326370
Adjacent sequences: A071065 A071066 A071067 * A071069 A071070 A071071


KEYWORD

easy,nonn


AUTHOR

Benoit Cloitre, May 26 2002


STATUS

approved



