OFFSET
1,3
COMMENTS
Conjecture: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 1, 2, 111, 127, 143, 158, 221, 223, 240, 460, 463, 480, 545, 560, 561, 1455, 1695, 1776, 2175. Moreover, any integer n > 10^4 not among 10543, 17935, 37583, 40383, 78543 can be written as u^4 + v^4 + 2*w^4 + 3*x^4 + 4*y^4 + 6*z^4 with u,v,w,x,y,z nonnegative integers.
If a(1),...,a(7) are positive integers with a(1) <= a(2) <= ... <= a(7) and a(1)+...+a(7) = g(4) = 19 such that {a(1)*x(1)^4+...+a(7)*x(7)^4: x(1),...,x(7) = 0,1,2,...} = {0,1,2,...}, then the tuple (a(1),...,a(7)) must be (1,1,2,2,3,4,6) or (1,1,2,2,3,3,7). Similarly, if a(1),...,a(8) are positive integers with a(1) <= a(2) <= ... <= a(8) and a(1)+...+a(8) = g(5) = 37 such that {a(1)*x(1)^5+...+a(8)*x(8)^5: x(1),...,x(8) = 0,1,2,...} = {0,1,2,...}, then (a(1),...,a(8)) must be (1,1,2,3,4,6,8,12) or (1,1,2,3,4,5,7,14).
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.
EXAMPLE
a(111) = 1 since 111 = 2*1 + 2^4 + 3^4 + 2*1^4 + 3*0^4 + 4*1^4 + 6*1^4.
a(240) = 1 since 240 = 2*0 + 2^4 + 2^4 + 2*0^4 + 3*2^4 + 4*2^4 + 6*2^4.
a(1776) = 1 since 1776 = 2*0 + 4^4 + 5^4 + 2*3^4 + 3*3^4 + 4*1^4 + 6*3^4.
a(2175) = 1 since 2175 = 2*1 + 0^4 + 4^4 + 2*2^4 + 3*5^4 + 4*1^4 + 6*1^4.
MATHEMATICA
QQ[n_]:=QQ[n]=n>0&&IntegerQ[n^(1/4)]
Do[r=0; Do[If[QQ[n-2t-6*z^4-4y^4-3x^4-2w^4-u^4], r=r+1], {t, 0, Min[1, n/2]}, {z, 0, ((n-2t^8)/6)^(1/4)}, {y, 0, ((n-2t-6z^4)/4)^(1/4)}, {x, 0, ((n-2t-6z^4-4y^4)/3)^(1/4)},
{w, 0, ((n-2t-6z^4-4y^4-3x^4)/2)^(1/4)}, {u, 0, ((n-2t-6z^4-4y^4-3x^4-2w^4)/2)^(1/4)}]; Print[n, " ", r]; Continue, {n, 1, 70}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Apr 07 2016
STATUS
approved