A260418 Number of ways to write 12*n+5 as 4*x^4 + 4*y^2 + z^2, where x is a nonnegative integer, and y and z are positive integers.

1, 2, 2, 2, 1, 3, 2, 4, 3, 2, 3, 2, 5, 2, 3, 3, 2, 4, 3, 3, 3, 2, 5, 2, 3, 3, 2, 6, 3, 4, 3, 5, 6, 3, 3, 5, 3, 5, 4, 2, 5, 4, 7, 3, 2, 7, 4, 6, 2, 2, 4, 3, 8, 4, 1, 2, 4, 8, 6, 2, 5, 2, 7, 4, 4, 3, 4, 5, 2, 4, 5, 6, 4, 3, 2, 5, 2, 7, 4, 5, 5, 2, 5, 3, 6, 5, 4, 7, 3, 4, 3, 5, 9, 3, 4, 3, 5, 11, 4, 5, 5

Offset

0,2

Comments

Conjecture: (i) a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 4, 54, 159, 289, 999, 1175, 1404, 16391, 39688.

(ii) All the numbers 24*n+4 (n = 0,1,2,...) can be written as 3*x^4+24*y^2+z^2, where x,y,z are integers with x > 0 and z > 0.

(iii) All the numbers 24*n+13 (n = 0,1,2,...) can be written as 3*x^4+9*y^2+z^2 with x,y,z positive integers.

(iv) All the numbers 24*n+r (n = 0,1,2,...) can be written as a*x^4+b*y^2+c*z^2 with x an integer and y and z positive integers, provided that (r,a,b,c) is among the following quadruples: (5,3,1,1), (5,12,4,1), (7,3,6,1), (10,5,1,1), (11,3,8,3), (11,6,3,2), (17,9,4,1).

See A290491 for a similar conjecture.

Links

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

Zhi-Wei Sun, On universal sums of polygonal numbers, Sci. China Math. 58(2015), 1367-1396.

Zhi-Wei Sun, On universal sums x(ax+b)/2+y(cy+d)/2+z(ez+f)/2, arXiv:1502.03056 [math.NT], 2015-2017.

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

Example

a(4) = 1 since 12*4+5 = 4*0^4 + 4*1^2 + 7^2.
a(54) = 1 since 12*54+5 = 4*0^4 + 4*11^2 + 13^2.
a(159) = 1 since 12*159+5 = 4*0^4 + 4*4^2 + 43^2.
a(289) = 1 since 12*289+5 = 4*1^4 + 4*19^2 + 45^2.
a(999) = 1 since 12*999+5 = 4*7^4 + 4*21^2 + 25^2.
a(1175) = 1 since 12*1175+5 = 4*3^4 + 4*55^2 + 41^2.
a(1404) = 1 since 12*1404+5 = 4*3^4 + 4*10^2 + 127^2.
a(16391) = 1 since 12*16391+5 = 4*5^4 + 4*207^2 + 151^2.
a(39688) = 1 since 12*39688+5 = 4*5^4 + 4*50^2 + 681^2.

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[r=0; Do[If[SQ[12n+5-4x^4-4y^2], r=r+1], {x, 0, ((12n+5)/4)^(1/4)}, {y, 1, Sqrt[(12n+5-4x^4)/4]}]; Print[n, " ", r], {n, 0, 100}]

Crossrefs

Cf. A000290, A000583, A270566, A286885, A286944, A287616, A290342, A290472, A290491.

Sequence in context: A275853 A324100 A127953 * A280919 A209266 A359930

Adjacent sequences: A260415 A260416 A260417 * A260419 A260420 A260421

Keyword

nonn

Author

Zhi-Wei Sun, Aug 04 2017

Status

approved