A343950 Number of ways to write n as x + y + z with x^2 + 4*y^2 + 5*z^2 a square, where x,y,z are positive integers with y or z a positive power of two.

0, 0, 0, 1, 1, 0, 0, 3, 1, 2, 2, 2, 3, 1, 4, 3, 2, 3, 3, 4, 4, 2, 1, 4, 6, 4, 2, 3, 12, 5, 3, 5, 8, 4, 5, 5, 8, 4, 7, 4, 4, 4, 7, 5, 5, 1, 4, 6, 5, 6, 6, 10, 7, 4, 9, 5, 10, 16, 7, 7, 9, 6, 5, 5, 14, 8, 6, 6, 3, 7, 1, 5, 4, 10, 5, 7, 10, 8, 13, 10, 3, 4, 8, 5, 12, 7, 20, 9, 12, 5, 8, 1, 9, 4, 8, 9, 8, 7, 4, 10

Offset

1,8

Comments

Conjecture 1: a(n) > 0 for all n > 7.

We have verified a(n) > 0 for all n = 8..50000. Clearly, a(2*n) > 0 if a(n) > 0.

Conjecture 2: For any integer n > 7, we can write n as x + y + z with x,y,z positive integers such that x^2 + 2*y^2 + 3*z^2 is a square.

Conjecture 3: For any integer n > 4, we can write n as x + y + z with x,y,z positive integers such that 3*x^2 + 4*y^2 + 5*z^2 (or x^2 + 3*y^2 + 5*z^2) is a square.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..3200

Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190. See also arXiv:1604.06723 [math.NT].

Example

a(4) = 1, and 4 = 1 + 1 + 2 with 1^2 + 4*1^2 + 5*2^2 = 5^2.
a(5) = 1, and 5 = 2 + 2 + 1 with 2^2 + 4*2^2 + 5*1^2 = 5^2.
a(9) = 1, and 9 = 4 + 1 + 4 with 4^2 + 4*1^2 + 5*4^2 = 10^2.
a(14) = 1, and 14 = 7 + 5 + 2 with 7^2 + 4*5^2 + 5*2^2 = 13^2.
a(23) = 1, and 23 = 7 + 8 + 8 with 7^2 + 4*8^2 + 5*8^2 = 25^2.
a(46) = 1, and 46 = 14 + 16 + 16 with 14^2 + 4*16^2 + 5*16^2 = 50^2.
a(71) = 1, and 71 = 42 + 8 + 21 with 42^2 + 4*8^2 + 5*21^2 = 65^2.
a(92) = 1, and 92 = 28 + 32 + 32 with 28^2 + 4*32^2 + 5*32^2 = 100^2.
a(142) = 1, and 142 = 84 + 16 + 42 with 84^2 + 4*16^2 + 5*42^2 = 130^2.

Mathematica

PowQ[n_]:=PowQ[n]=n>1&&IntegerQ[Log[2, n]];
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={}; Do[r=0; Do[If[(PowQ[y]||PowQ[n-x-y])&&SQ[x^2+4*y^2+5*(n-x-y)^2], r=r+1], {x, 1, n-3}, {y, 1, n-1-x}]; tab=Append[tab, r], {n, 1, 100}]; Print[tab]

Crossrefs

Cf. A000079, A000290, A271510, A271513, A271518, A230121, A230747.

Sequence in context: A029418 A185736 A144148 * A085247 A003016 A328848

Adjacent sequences: A343947 A343948 A343949 * A343951 A343952 A343953

Keyword

nonn

Author

Zhi-Wei Sun, May 05 2021

Status

approved