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

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A303338 Number of ways to write n as x^2 + 2*y^2 + 3*2^z + 4^w with x,y,z,w nonnegative integers. 32
0, 0, 0, 1, 1, 1, 3, 3, 2, 4, 3, 2, 6, 2, 4, 8, 2, 4, 7, 3, 4, 8, 5, 5, 10, 6, 4, 10, 8, 5, 12, 7, 3, 12, 4, 5, 12, 5, 5, 14, 7, 4, 12, 7, 6, 12, 6, 6, 10, 7, 7, 12, 7, 6, 14, 6, 8, 16, 4, 8, 18, 5, 6, 16, 5, 9, 13, 7, 7, 14 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,7

COMMENTS

Conjecture: a(n) > 0 for all n > 3.

This is stronger than the author's previous conjecture in A302983. I call it the 1-2-3-4 conjecture. It has been verified that a(n) > 0 for all n = 4..10^9.

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, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.

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

EXAMPLE

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

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

a(6) = 1 with 6 = 0^2 + 2*1^2 + 3*2^0 + 4^0.

a(9) = 2 with 9 = 0^2 + 2*1^2 + 3*2^0 + 4^1 = 0^2 + 2*1^2 + 3*2^1 + 4^0.

MATHEMATICA

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

g[n_]:=g[n]=Sum[Boole[MemberQ[{5, 7}, Mod[Part[Part[f[n], i], 1], 8]]&&Mod[Part[Part[f[n], i], 2], 2]==1], {i, 1, Length[f[n]]}]==0;

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

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

tab={}; Do[r=0; Do[If[QQ[n-3*2^k-4^j], Do[If[SQ[n-3*2^k-4^j-2x^2], r=r+1], {x, 0, Sqrt[(n-3*2^k-4^j)/2]}]], {k, 0, Log[2, n/3]}, {j, 0, If[3*2^k==n, -1, Log[4, n-3*2^k]]}]; tab=Append[tab, r], {n, 1, 70}]; Print[tab]

CROSSREFS

Cf. A000079, A000290, A002479, A271518, A281976, A299924, A299537, A299794, A300219, A300362, A300396, A300441, A301376, A301391, A301471, A301472, A302920, A302981, A302982, A302983, A302984, A302985, A303363.

Sequence in context: A247509 A106686 A106702 * A271510 A282545 A306471

Adjacent sequences:  A303335 A303336 A303337 * A303339 A303340 A303341

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Apr 22 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 March 18 22:11 EDT 2019. Contains 321305 sequences. (Running on oeis4.)