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!)
A301452 Number of ways to write n^2 as m*4^k + x^2 + 2*y^2 with m in the set {2, 3} and k,x,y nonnegative integers. 11
0, 2, 2, 2, 2, 5, 3, 2, 4, 4, 4, 5, 5, 5, 6, 2, 4, 6, 5, 4, 9, 5, 4, 5, 5, 7, 10, 5, 6, 7, 8, 2, 6, 6, 7, 6, 9, 7, 10, 4, 6, 12, 3, 5, 10, 5, 6, 5, 5, 8, 9, 7, 7, 12, 5, 5, 13, 9, 6, 7, 8, 10, 13, 2, 6, 8, 10, 6, 15, 9, 9, 6, 10, 9, 12, 7, 8, 13, 6, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Conjecture: a(n) > 0 for all n > 1.

We call this the 2-3 conjecture. It is simialr to the author's 2-5 conjecture which states that A300510(n) > 0 for all n > 1.

We have verified that a(n) > 0 for all n = 2..5*10^7.

It is known that the number of ways to write a positive integer n as x^2 + 2*y^2 with x and y integers is twice the difference |{d > 0: d|n and d == 1,3 (mod 8)| - |{d>0: d|n and d == 5,7 (mod 8)}|.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.

Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

EXAMPLE

a(2) = 2 since 2^2 = 2*4^0 + 0^2 + 2*1^2 and 2^2 = 3*4^0 + 1^2 + 2*0^2.

a(3) = 2 since 3^2 = 2*4^1 + 1^2 + 2*0^2 and 3^2 = 3*4^0 + 2^2 + 2*1^2.

a(5) = 2 since 5^2 = 2*4^1 + 3^2 + 2*2^2 and 5^2 = 3*4^0 + 2^2 + 2*3^2.

MATHEMATICA

f[n_]:=f[n]=n/2^(IntegerExponent[n, 2]);

OD[n_]:=OD[n]=Divisors[f[n]];

QQ[n_]:=QQ[n]=(n==0)||(n>0&&Sum[JacobiSymbol[-2, Part[OD[n], i]], {i, 1, Length[OD[n]]}]!=0);

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];

tab={}; Do[r=0; Do[If[QQ[n^2-m*4^k], Do[If[SQ[n^2-m*4^k-2x^2], r=r+1], {x, 0, Sqrt[(n^2-m*4^k)/2]}]], {m, 2, 3}, {k, 0, Log[4, n^2/m]}]; tab=Append[tab, r], {n, 1, 80}]; Print[tab]

CROSSREFS

Cf. A000290, A000302, A002479, A299924, A299537, A299794, A300219, A300362, A300396, A300510, A301376, A301391.

Sequence in context: A080647 A324516 A181058 * A322788 A177333 A118486

Adjacent sequences:  A301449 A301450 A301451 * A301453 A301454 A301455

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Mar 21 2018

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 14 19:27 EST 2019. Contains 329987 sequences. (Running on oeis4.)