login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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 (list; graph; refs; listen; history; text; internal format)
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(n-k+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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 6 19:31 EST 2019. Contains 329809 sequences. (Running on oeis4.)