

A085263


Number of ways to write n as the sum of a squarefree number (A005117) and a square (A000290).


5



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, 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, 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
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OFFSET

1,6


COMMENTS

a(A085265(n))>0; a(A085266(n))=1; a(A085267(n))>1.
a(A085264(n))=n and a(i)<>n for i < A085264(n).
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


LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Square Numbers.
Eric Weisstein's World of Mathematics, Squarefree
Robert G. Wilson v, Plot of first 100000 terms


FORMULA

a(n+1) = SUM(A008966(k)*A010052(nk+1): 1<=k<=n).  Reinhard Zumkeller, Nov 04 2009
a(n) < sqrt(n).  Robert G. Wilson v, May 17 2014
G.f.: (Sum_{i>=1} x^(i^2))*(Sum_{j>=1} mu(j)^2*x^j).  Ilya Gutkovskiy, Feb 06 2017


EXAMPLE

a(11)=3:
11 = 1 + 10 = A000290(1) + A005117(7)
= 4 + 7 = A000290(2) + A005117(6)
= 9 + 2 = A000290(3) + A005117(2).


MATHEMATICA

f[n_] := Count[ SquareFreeQ@# & /@ (n  Range[1, Floor[ Sqrt[ n]]]^2), True]; Array[f, 105] (* Robert G. Wilson v, May 16 2014 *)


CROSSREFS

Cf. A002636, A115288, A160324, A160325, A160326, A240088, A242442, A242443.
Sequence in context: A098281 A207324 A103343 * A115092 A172281 A304945
Adjacent sequences: A085260 A085261 A085262 * A085264 A085265 A085266


KEYWORD

nonn,look


AUTHOR

Reinhard Zumkeller, Jun 23 2003


STATUS

approved



