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A290491 Number of ways to write 12*n+1 as x^2 + 4*y^2 + 8*z^4, where x and y are positive integers and z is a nonnegative integer. 4
2, 2, 2, 1, 2, 3, 2, 2, 2, 1, 3, 4, 3, 2, 3, 5, 3, 1, 4, 3, 3, 4, 3, 2, 2, 5, 5, 1, 3, 2, 4, 3, 3, 3, 2, 6, 3, 2, 1, 3, 6, 4, 2, 2, 3, 5, 3, 3, 1, 2, 4, 3, 4, 2, 5, 4, 5, 5, 3, 2, 7, 4, 4, 3, 2, 6, 3, 4, 4, 3, 9, 3, 2, 3, 3, 6, 5, 4, 4, 3, 8, 5, 2, 2, 3, 6, 3, 3, 4, 4, 5, 7, 2, 3, 3, 8, 6, 1, 5, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 4, 10, 18, 28, 39, 49, 98, 142, 163, 184, 208, 320, 382, 408, 814, 910, 1414, 2139, 2674, 3188, 3213, 4230, 6279, 25482.

(ii) All the numbers 16*n+5 (n = 0,1,2,...) can be written as x^4 + 4*y^2 + z^2, where x,y,z are integers with y > 0 and z > 0.

(iii) All the numbers 24*n+1 (n = 0,1,2,...) can be written as 12*x^4 + 4*y^2 + z^2 with x,y,z integers. Also, all the numbers 24*n+9 (n = 0,1,2,...) can be written as 2*x^4 + 6*y^2 + z^2 with x,y,z positive integers.

(iv) All the numbers 24*n+2 (n = 0,1,2,...) can be written as x^4 + 9*y^2 + z^2, where x,y,z are integers with z > 0. Also, all the numbers 24*n+17 (n = 0,1,2,...) can be written as x^4 + 16*y^2 + z^2, where x,y,z are integers with y > 0 and z > 0.

(v) All the numbers 30*n+3 (n = 1,2,3,...) can be written as 2*x^4 + 3*y^2 + z^2 with x,y,z positive integers. Also, all the numbers 30*n+21 (n = 0,1,2,...) can be written as 2*x^4 + 3*y^2 + z^2, where x,y,z are integers with z > 0.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..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(10) = 1 since 12*10+1 = 7^2 + 4*4^2 + 8*1^4.

a(28) = 1 since 12*28+1 = 9^2 + 4*8^2 + 8*0^4.

a(49) = 1 since 12*49+1 = 19^2 + 4*5^2 + 8*2^4.

a(3188) = 1 since 12*3188+1 = 103^2 + 4*80^2 + 8*4^4.

a(3213) = 1 since 12*3213+1 = 91^2 + 4*87^2 + 8*0^4.

a(4230) = 1 since 12*4230+1 = 223^2 + 4*16^2 + 8*1^4.

a(6279) = 1 since 12*6279+1 = 19^2 + 4*75^2 + 8*9^4.

a(25482) = 1 since 12*25482+1 = 531^2 + 4*58^2 + 8*6^4.

MATHEMATICA

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];

Do[r=0; Do[If[SQ[12n+1-8x^4-4y^2], r=r+1], {x, 0, ((12n+1)/8)^(1/4)}, {y, 1, Sqrt[(12n+1-8x^4)/4]}]; Print[n, " ", r], {n, 1, 100}]

CROSSREFS

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

Sequence in context: A185636 A333212 A182597 * A194314 A006371 A000177

Adjacent sequences:  A290488 A290489 A290490 * A290492 A290493 A290494

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Aug 03 2017

STATUS

approved

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Last modified June 1 04:57 EDT 2020. Contains 334758 sequences. (Running on oeis4.)