

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
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OFFSET

1,2


COMMENTS

Conjecture: a(n) > 0 for all n > 1.
We call this the 23 conjecture. It is simialr to the author's 25 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: dn and d == 1,3 (mod 8)  {d>0: dn and d == 5,7 (mod 8)}.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000
ZhiWei Sun, Refining Lagrange's foursquare theorem, J. Number Theory 175(2017), 167190.
ZhiWei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 20172018.


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^2m*4^k], Do[If[SQ[n^2m*4^k2x^2], r=r+1], {x, 0, Sqrt[(n^2m*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

ZhiWei Sun, Mar 21 2018


STATUS

approved



