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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 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A033265 A096004 A193495 * A352104 A240872 A328806
KEYWORD
easy,nonn
AUTHOR
Benoit Cloitre, May 26 2002
STATUS
approved

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Last modified July 21 06:08 EDT 2024. Contains 374463 sequences. (Running on oeis4.)