login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A352627 Number of ways to write n as a^2 + 2*b^2 + c^4 + 4*d^4 + c^2*d^2, where a,b,c,d are nonnegative integers. 5
1, 2, 2, 2, 3, 2, 3, 3, 3, 4, 4, 1, 4, 3, 1, 3, 3, 4, 5, 4, 3, 1, 5, 3, 5, 6, 3, 4, 6, 1, 2, 3, 3, 8, 5, 3, 4, 4, 4, 3, 5, 3, 6, 4, 3, 2, 1, 2, 4, 6, 4, 5, 5, 1, 5, 5, 2, 7, 5, 2, 6, 2, 1, 3, 3, 5, 4, 7, 5, 2, 7, 2, 8, 10, 3, 6, 5, 3, 6, 2, 4, 9, 10, 6, 3, 5, 4, 8, 7, 6, 6, 5, 5, 3, 3, 2, 8, 11, 7, 9, 11 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Conjecture: a(n) > 0 for all n = 0,1,2,.... In other words, each nonnegative integer can be written as a^2 + 2*b^2 + c^4 + 4*d^4 + c^2*d^2 with a,b,c,d integers.
It seems that a(n) = 1 only for n = 0, 11, 14, 21, 29, 46, 53, 62, 149, 174, 221, 239, 254, 1039, 1709, 2239.
See also A352628, A352629 and A352632 for similar conjectures.
LINKS
EXAMPLE
a(11) = 1 with 11 = 3^2 + 2*1^2 + 0^4 + 4*0^4 + 0^2*0^2.
a(14) = 1 with 14 = 0^2 + 2*2^2 + 1^4 + 4*1^4 + 1^2*1^2.
a(221) = 1 with 221 = 12^2 + 2*2^2 + 1^4 + 4*2^4 + 1^2*2^2.
a(239) = 1 with 239 = 15^2 + 2*2^2 + 1^4 + 4*1^4 + 1^2*1^2.
a(254) = 1 with 254 = 1^2 + 2*6^2 + 3^4 + 4*2^4 + 3^2*2^2.
a(1039) = 1 with 1039 = 31^2 + 2*6^2 + 1^4 + 4*1^4 + 1^2*1^2.
a(1709) = 1 with 1709 = 9^2 + 2*26^2 + 4^4 + 4*1^4 + 4^2*1^2.
a(2239) = 1 with 2239 = 41^2 + 2*6^2 + 3^4 + 4*3^4 + 3^2*3^2.
MATHEMATICA
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={}; Do[r=0; Do[If[SQ[n-4d^4-c^4-c^2*d^2-2b^2], r=r+1], {d, 0, (n/4)^(1/4)}, {c, 0, Sqrt[(Sqrt[4n-15*d^4]-d^2)/2]}, {b, 0, Sqrt[(n-4d^4-c^4-c^2*d^2)/2]}]; tab=Append[tab, r], {n, 0, 100}]; Print[tab]
CROSSREFS
Sequence in context: A342739 A137734 A366066 * A182210 A078705 A346018
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Mar 24 2022
STATUS
approved

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

License Agreements, Terms of Use, Privacy Policy. .

Last modified February 23 04:22 EST 2024. Contains 370266 sequences. (Running on oeis4.)