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A351206
Least positive integer m such that n = x^4 + (y^4 + z^4 + 7*w^2)/m^4 for some nonnegative integers x,y,z,w with y <= z.
1
1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 4, 3, 1, 1, 1, 6, 2, 2, 10, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 3, 4, 2, 4, 10, 1, 1, 2, 4, 2, 1, 1, 1, 3, 1, 3, 2, 2, 2, 2, 2, 1, 3, 4, 2, 2, 1, 1, 3, 1, 1, 1, 1, 2, 4, 2, 2, 3, 8, 10, 2, 2, 1, 10, 5, 1, 1, 1, 1, 1, 4, 2, 2, 8, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 4, 4
OFFSET
0,5
COMMENTS
Conjecture: a(n) exists for any nonnegative integer n.
This implies that each nonnegative rational number can be written as 7*w^2 + x^4 + y^4 + z^4 with w,x,y,z rational numbers.
LINKS
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34 (2017), no.2, 97-120.
Zhi-Wei Sun, Sums of four rational squares with certain restrictions, arXiv:2010.05775 [math.NT], 2020-2022.
EXAMPLE
a(6) = 2 with 6 = 1^4 + (1^4 + 2^4 + 7*3^2)/2^4.
a(19) = 6 with 19 = 0^4 + (1^4 + 4^4 + 7*59^2)/6^4.
a(22) = 10 with 22 = 2^4 + (2^4 + 13^4 + 7*67^2)/10^4.
a(5797) = 20 with 5797 = 0^4 + (81^4 + 164^4 + 7*4797^2)/20^4.
MATHEMATICA
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={}; Do[m=1; Label[bb]; k=m^4; Do[If[SQ[(k*(n-x^4)-y^4-z^4)/7], tab=Append[tab, m]; Goto[aa]], {x, 0, n^(1/4)}, {y, 0, (k*(n-x^4)/2)^(1/4)}, {z, y, (k*(n-x^4)-y^4)^(1/4)}]; m=m+1; Goto[bb]; Label[aa], {n, 0, 100}]; Print[tab]
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Feb 04 2022
STATUS
approved