A352503 Number of ways to write n as w^3 + 2*x^3 + 4*y^3 + 5*z^3 + t^6, where w is a positive integer, and x,y,z,t are nonnegative integers.

1, 1, 1, 1, 1, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 3, 2, 1, 2, 4, 4, 2, 2, 1, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 5, 3, 2, 1, 2, 2, 2, 3, 2, 2, 1, 2, 3, 4, 4, 1, 4, 5, 3, 6, 4, 5, 4, 5, 5, 3, 5, 3, 5, 1, 1, 1, 3, 6, 2, 3, 2, 4, 4, 3, 3, 2, 4, 2, 2, 3, 1, 3, 4, 5, 2, 5, 4

Offset

1,6

Comments

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

This has been verified for n = 1..10^6.

It seems that a(n) = 1 only for n = 1..5, 16, 19, 20, 21, 23, 24, 25, 26, 31, 37, 51, 58, 63, 77, 78, 79, 94, 108, 207, 208, 218, 316, 487, 490, 559.

Links

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

Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34 (2017), no. 2, 97-120. (See Conjecture 3.4(i).)

Example

a(20) = 1 with 20 = 2^3 + 2*1^3 + 4*1^3 + 5*1^3 + 1^6.
a(79) = 1 with 79 = 2^3 + 2*1^3 + 4*0^3 + 5*1^3 + 2^6.
a(316) = 1 with 316 = 1^3 + 2*3^3 + 4*4^3 + 5*1^3 + 0^6.
a(487) = 1 with 487 = 5^3 + 2*!^3 + 4*4^3 + 5*2^3 + 2^6.
a(490) = 1 with 490 = 2^3 + 2*3^3 + 4*3^3 + 5*4^3 + 0^6.
a(559) = 1 with 559 = 8^3 + 2*1^3 + 4*1^3 + 5*2^3 + 1^6.

Mathematica

CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)];
tab={}; Do[r=0; Do[If[CQ[n-t^6-2x^3-4y^3-5z^3], r=r+1], {t, 0, (n-1)^(1/6)}, {x, 0, ((n-1-t^6)/2)^(1/3)}, {y, 0, ((n-1-t^6-2x^3)/4)^(1/3)}, {z, 0, ((n-1-t^6-2x^3-4y^3)/5)^(1/3)}]; tab=Append[tab, r], {n, 1, 100}]; Print[tab]

Crossrefs

Cf. A000578, A001014, A271099, A271237.

Sequence in context: A375760 A145989 A117183 * A071187 A329614 A134852

Adjacent sequences: A352500 A352501 A352502 * A352504 A352505 A352506

Keyword

nonn

Author

Zhi-Wei Sun, Mar 28 2022

Status

approved