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

Author

Zhi-Wei Sun, Mar 28 2022

Keywords

• nonn

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.

Examples

			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.
		

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).)

Crossrefs

Cf. A000578, A001014, A271099, A271237.

Programs

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]