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A300448 Number of the integers 4^k*m with k >= 0 and m = 1, 2, 5 such that n^2 - 4^k*m can be written as the sum of two squares. 2
1, 2, 4, 2, 4, 5, 4, 2, 7, 5, 6, 5, 6, 5, 5, 2, 5, 7, 6, 5, 10, 7, 5, 5, 7, 7, 6, 5, 6, 6, 7, 2, 10, 6, 8, 7, 7, 6, 6, 5, 7, 11, 5, 7, 10, 5, 6, 5, 8, 8, 9, 7, 8, 6, 6, 5, 10, 7, 5, 6, 5, 8, 7, 2, 5, 10, 8, 6, 10, 8, 7, 7, 11, 7, 7, 6, 10, 7, 5, 5 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

Conjecture 2: Each positive square n^2 can be written as 4^k*m + x^2 + y^2 with k,x,y nonnegative integers and m among 1, 5 17.

Conjecture 3: For each n = 1,2,3,... we can write n^2 as 4^k*m + x^2 + y^2 with k,x,y nonnegative integers and m among 1, 5, 65.

Conjecture 3 implies that A300441(n) > 0 for all n > 0 since  4*A001353(0)^2 + 1 = 1, 4*A001353(1)^2 + 1 = 5 and 4*A001353(2)^2 + 1 = 65.

We have verified Conjectures 1-3 for all n = 1..2*10^7.

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(1) = 1 since 1^2 - 4^0*1 = 0 = 0^2 + 0^2.

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

a(4) = 2 since 4^2 - 4^1*2 = 8 = 2^2 + 2^2 and 4^2 - 4^2*1 = 0 = 0^2 + 0^2.

MATHEMATICA

f[n_]:=f[n]=FactorInteger[n];

g[n_]:=g[n]=Sum[If[Mod[Part[Part[f[n], i], 1]-3, 4]==0&&Mod[Part[Part[f[n], i], 2], 2]==1, 1, 0], {i, 1, Length[f[n]]}]==0;

QQ[n_]:=QQ[n]=n==0||(n>0&&g[n]);

u[1]=1; u[2]=2; u[3]=5;

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

CROSSREFS

Cf. A000118, A000302, A001481, A271518, A281976, A299924, A299537, A299794, A300362, A300441, A300396.

Sequence in context: A018838 A116982 A216621 * A214781 A214850 A236186

Adjacent sequences:  A300445 A300446 A300447 * A300449 A300450 A300451

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Mar 06 2018

STATUS

approved

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Last modified April 22 19:11 EDT 2021. Contains 343177 sequences. (Running on oeis4.)