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%I #30 Oct 31 2020 13:00:02
%S 0,1,1,1,1,2,2,1,1,2,3,2,0,3,3,2,2,3,3,2,2,3,4,2,1,4,4,2,1,5,4,3,2,2,
%T 5,2,3,6,6,3,2,6,4,3,2,5,6,3,2,5,6,3,2,4,6,4,3,4,6,4,1,7,5,3,3,7,6,4,
%U 4,6,8,3,3,6,7,2,4,8,5,4,3,7,9,4,2,8,9,4,3,6,6,5,4,7,9,5,3,8,4,3,5,9
%N Number of ways to write n as the sum of a squarefree number (A005117) and a positive square (A000290).
%C a(A085265(n))>0; a(A085266(n))=1; a(A085267(n))>1.
%C a(A085264(n))=n and a(i)<>n for i < A085264(n).
%C First occurrence of k: 2, 6, 11, 23, 30, 38, 62, 71, 83, 110, 138, 155, 182, 203, 227, 263, 302, 327, 383, 435, 447, 503, 542, 602, 635, ..., . Conjecture: For each k above, there is a finite number of terms; for example, only the two numbers 1 and 13 cannot be represented as the sum of a squarefree number and a square. The number of k terms beginning with 0: 2, 9, 19, 27, 38, 36, 57, 63, 62, 74, 94, ..., . - _Robert G. Wilson v_, May 16 2014
%H T. D. Noe, <a href="/A085263/b085263.txt">Table of n, a(n) for n = 1..10000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SquareNumber.html">Square Numbers.</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Squarefree.html">Squarefree</a>
%H Robert G. Wilson v, <a href="/A085263/a085263.pdf">Plot of first 100000 terms</a>
%F a(n+1) = Sum_{k=1..n} A008966(k)*A010052(n-k+1). - _Reinhard Zumkeller_, Nov 04 2009
%F a(n) < sqrt(n). - _Robert G. Wilson v_, May 17 2014
%F G.f.: (Sum_{i>=1} x^(i^2))*(Sum_{j>=1} mu(j)^2*x^j). - _Ilya Gutkovskiy_, Feb 06 2017
%e a(11)=3:
%e 11 = 1 + 10 = A000290(1) + A005117(7)
%e = 4 + 7 = A000290(2) + A005117(6)
%e = 9 + 2 = A000290(3) + A005117(2).
%t f[n_] := Count[ SquareFreeQ@# & /@ (n - Range[1, Floor[ Sqrt[ n]]]^2), True]; Array[f, 105] (* _Robert G. Wilson v_, May 16 2014 *)
%o (PARI) a(n) = sum(k=1, n-1, issquare(k) * issquarefree(n-k)); \\ _Michel Marcus_, Oct 30 2020
%Y Cf. A000290, A002636, A005117, A115288, A160324, A160325, A160326, A240088, A242442, A242443.
%K nonn,look
%O 1,6
%A _Reinhard Zumkeller_, Jun 23 2003
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