

A271518


Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with x + 3*y + 5*z a square, where w, x, y and z are nonnegative integers.


141



1, 2, 2, 2, 2, 1, 1, 1, 1, 3, 3, 2, 2, 2, 4, 2, 2, 5, 5, 3, 2, 2, 2, 3, 1, 5, 5, 2, 2, 5, 8, 1, 2, 6, 3, 3, 2, 3, 7, 5, 2, 8, 6, 1, 4, 6, 6, 2, 2, 6, 9, 5, 4, 3, 7, 6, 2, 6, 7, 5, 2, 1, 6, 6, 2, 10, 9, 6, 3, 3, 6, 2, 3, 8, 12, 5, 5, 7, 11, 5, 1
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OFFSET

0,2


COMMENTS

Conjecture: (i) a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 4^k*6 (k = 0,1,2,...), 16^k*m (k = 0,1,2,... and m = 5, 7, 8, 31, 43, 61, 116).
(ii) Any integer n > 15 can be written as w^2 + x^2 + y^2 + z^2 with w, x, y, z nonnegative integers and 6*x + 10*y + 12*z a square.
(iii) Each nonnegative integer n not among 7, 15, 23, 71, 97 can be written as w^2 + x^2 + y^2 + z^2 with w, x, y, z nonnegative integers and 2*x + 6*y + 10*z a square. Also, any nonnegative integer n not among 7, 43, 79 can be written as w^2 + x^2 + y^2 + z^2 with w, x, y, z nonnegative integers and 3*x + 5*y + 6*z a square.
See also A271510 and A271513 for related conjectures.
a(n) > 0 verified for all n <= 3*10^7.  ZhiWei Sun, Nov 28 2016
QingHu Hou at Tianjin Univ. has verified a(n) > 0 and parts (ii) and (iii) of the above conjecture for n up to 10^9.  ZhiWei Sun, Dec 04 2016
The conjecture that a(n) > 0 for all n = 0,1,2,... is called the 135Conjecture and the author has announced a prize of 1350 US dollars for its solution.  ZhiWei Sun, Jan 17 2017
QingHu Hou has finished his verification of a(n) > 0 for n up to 10^10.  ZhiWei Sun, Feb 17 2017
The 135 conjecture was finally proved by António Machiavelo and Nikolaos Tsopanidis in a JNT paper published in 2021. This is a great achivement!  ZhiWei Sun, Mar 31 2021


LINKS

ZhiWei Sun, Table of n, a(n) for n = 0..10000
António Machiavelo and Nikolaos Tsopanidis, ZhiWei Sun's 135 Conjecture and Variations, arXiv:2003.02592 [math.NT], 2020.
António Machiavelo and Nikolaos Tsopanidis, ZhiWei Sun's 135 Conjecture and Variations, J. Number Theory 222 (2021), 120.
António Machiavelo, Rogério Reis, and Nikolaos Tsopanidis, Report on ZhiWei Sun's "135 conjecture" and some of its refinements, arXiv:2005.13526 [math.NT], 2020.
António Machiavelo, Rogério Reis, and Nikolaos Tsopanidis, Report on ZhiWei Sun's 135 conjecture and some of its refinements, J. Number Theory 222 (2021), 2129.
ZhiWei Sun, Refining Lagrange's foursquare theorem, arXiv:1604.06723 [math.NT], 2016.
ZhiWei Sun, Refining Lagrange's foursquare theorem, J. Number Theory 175(2017), 167190. (See Conjecture 4.3(i) and Remark 4.3.)


EXAMPLE

a(5) = 1 since 5 = 2^2 + 1^2 + 0^2 + 0^2 with 1 + 3*0 + 5*0 = 1^2.
a(6) = 1 since 6 = 2^2 + 1^2 + 1^2 + 0^2 with 1 + 3*1 + 5*0 = 2^2.
a(7) = 1 since 7 = 2^2 + 1^2 + 1^2 + 1^2 with 1 + 3*1 + 5*1 = 3^2.
a(8) = 1 since 8 = 0^2 + 0^2 + 2^2 + 2^2 with 0 + 3*2 + 5*2 = 4^2.
a(24) = 1 since 24 = 4^2 + 0^2 + 2^2 + 2^2 with 0 + 3*2 + 5*2 = 4^2.
a(31) = 1 since 31 = 1^2 + 5^2 + 2^2 + 1^2 with 5 + 3*2 + 5*1 = 4^2.
a(43) = 1 since 43 = 1^2 + 1^2 + 5^2 + 4^2 with 1 + 3*5 + 5*4 = 6^2.
a(61) = 1 since 61 = 6^2 + 0^2 + 0^2 + 5^2 with 0 + 3*0 + 5*5 = 5^2.
a(116) = 1 since 116 = 10^2 + 4^2 + 0^2 + 0^2 with 4 + 3*0 + 5*0 = 2^2.


MATHEMATICA

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0; Do[If[SQ[nx^2y^2z^2]&&SQ[x+3y+5z], r=r+1], {x, 0, Sqrt[n]}, {y, 0, Sqrt[nx^2]}, {z, 0, Sqrt[nx^2y^2]}]; Print[n, " ", r]; Continue, {n, 0, 80}]


CROSSREFS

Cf. A000118, A000290, A270969, A271510, A271513, A273294, A273302, A276533, A278560.
Sequence in context: A327051 A275301 A282542 * A352629 A106825 A156608
Adjacent sequences: A271515 A271516 A271517 * A271519 A271520 A271521


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Apr 09 2016


STATUS

approved



