A349246 Number of ways to write n as w^8 + x^4 + 2*y^4 + 4*z^4 + t*(t+1), where w, x, y, z, and t are nonnegative integers.

1, 2, 3, 4, 4, 4, 5, 6, 5, 4, 3, 2, 3, 4, 3, 2, 3, 3, 4, 4, 4, 4, 5, 5, 4, 4, 3, 3, 3, 2, 2, 3, 4, 5, 5, 5, 5, 5, 5, 5, 3, 1, 3, 5, 4, 4, 4, 3, 5, 6, 4, 2, 3, 4, 3, 2, 2, 4, 5, 4, 4, 4, 4, 5, 5, 4, 4, 6, 5, 3, 2, 2, 5, 6, 5, 5, 5, 5, 7, 8, 4, 2, 4, 5, 4, 5, 5, 6, 7, 6, 6, 5, 6, 8, 9, 8, 6, 7, 5, 3, 3

Offset

0,2

Comments

Conjecture: a(n) > 0 for all n = 0,1,2,....

This has been verified for all n = 0..10^8.

It seems that a(n) = 1 only for n = 0, 41, 131, 141, 145, 225, 251, 297, 591, 621, 916, 1021, 1241, 1431, 2025, 4691.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000

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

Example

a(145) = 1 with 145 = 0^8 + 3^4 + 2*0^4 + 4*2^4 + 0*1.
a(225) = 1 with 225 = 1^8 + 2^4 + 2*3^4 + 4*1^4 + 6*7.
a(916) = 1 with 916 = 2^8 + 2^4 + 2*4^4 + 4*0^4 + 11*12.
a(1021) = 1 with 1021 = 0^8 + 5^4 + 2*0^4 + 4*3^4 + 8*9.
a(1241) = 1 with 1241 = 0^8 + 5^4 + 2*0^4 + 4*2^4 + 23*24.
a(1431) = 1 with 1431 = 1^8 + 6^4 + 2*1^4 + 4*0^4 + 11*12.
a(2025) = 1 with 2025 = 2^8 + 3^4 + 2*2^4 + 4*3^4 + 36*37.
a(4691) = 1 with 4691 = 2^8 + 3^4 + 2*0^4 + 4*2^4 + 65*66.

Mathematica

QQ[n_]:=QQ[n]=IntegerQ[Sqrt[4n+1]];
tab={}; Do[r=0; Do[If[QQ[n-w^8-4z^4-2y^4-x^4], r=r+1], {w, 0, n^(1/8)}, {z, 0, ((n-w^8)/4)^(1/4)}, {y, 0, ((n-w^8-4z^4)/2)^(1/4)}, {x, 0, (n-w^8-4z^4-2y^4)^(1/4)}]; tab=Append[tab, r], {n, 0, 100}]; Print[tab]

Crossrefs

Cf. A000583, A001016, A002378.

Sequence in context: A060740 A307467 A145339 * A123273 A284359 A167991

Adjacent sequences: A349243 A349244 A349245 * A349247 A349248 A349249

Keyword

nonn

Author

Zhi-Wei Sun, Mar 26 2022

Status

approved