A351617 Number of ways to write n as 11^w + x^2 + 2*y^2 + 3*z^2 + x*y*z, where w,x,y,z are nonnegative integers.

1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 4, 4, 3, 4, 1, 3, 5, 3, 5, 1, 5, 5, 1, 3, 4, 3, 6, 5, 5, 2, 4, 4, 3, 2, 8, 4, 5, 5, 5, 2, 4, 3, 5, 3, 5, 5, 5, 5, 7, 3, 5, 5, 4, 4, 3, 4, 8, 3, 8, 2, 6, 8, 3, 5, 4, 5, 10, 1, 5, 1, 4, 7, 4, 4, 7, 8, 11, 1, 3, 4, 5, 6, 7, 5, 6, 7, 7, 1, 5, 4, 10, 4, 7, 7, 4, 3, 7, 3, 8

Offset

1,4

Comments

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

(ii) Let c be among 3, 4, 5, 7, 8. Then each positive integer n can be written as c^w + x^2 + 2*y^2 + 3*z^2 + x*y*z, where w,x,y,z are nonnegative integers.

This has been verified for all n = 1..3*10^5.

Links

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

Example

a(6) = 1 with 6 = 11^0 + 0^2 + 2*1^2 + 3*1^2 + 0*1*1.
a(24) = 1 with 24 = 11^1 + 1^2 + 2*0^2 + 3*2^2 + 1*0*2.
a(71) = 1 with 71 = 11^0 + 4^2 + 2*3^2 + 3*2^2 + 4*3*2.
a(89) = 1 with 89 = 11^0 + 4^2 + 2*6^2 + 3*0^2 + 4*6*0.
a(107) = 1 with 107 = 11^1 + 8^2 + 2*4^2 + 3*0^2 + 8*4*0.

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={}; Do[r=0; Do[If[SQ[4(n-11^w-2y^2-3z^2)+y^2*z^2], r=r+1], {w, 0, Log[11, n]}, {z, 0, Sqrt[(n-11^w)/3]}, {y, 0, Sqrt[(n-11^w-3z^2)/2]}]; tab=Append[tab, r], {n, 1, 100}]; Print[tab]

Crossrefs

Cf. A000290, A001014, A351723, A351902, A352259, A352286.

Sequence in context: A143331 A167677 A278387 * A074293 A013949 A331349

Adjacent sequences: A351614 A351615 A351616 * A351618 A351619 A351620

Keyword

nonn

Author

Zhi-Wei Sun, Mar 10 2022

Status

approved