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A271169
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Number of ordered ways to write n as s^5 + t^5 + 2*u^5 + 3*v^5 + 4*w^5 + 5*x^5 + 7*y^5 + 14*z^5, where s,t,u,v,w,x,y,z are nonnegative integers with s <= t.
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4
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1, 1, 2, 2, 3, 4, 4, 6, 5, 7, 6, 7, 7, 6, 8, 6, 8, 6, 7, 7, 6, 8, 6, 8, 6, 7, 7, 6, 7, 5, 6, 4, 5, 4, 3, 4, 3, 4, 3, 4, 4, 4, 5, 4, 5, 4, 5, 5, 4, 5, 4, 5, 4, 5, 5, 4, 5, 4, 5, 4, 4, 4, 3, 3, 4, 3, 3, 3, 4, 5, 3, 6, 4, 7, 5, 5, 7, 4, 8, 4, 7
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OFFSET
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0,3
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COMMENTS
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Conjecture: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 1, 2602.
Note that 1+1+2+3+4+5+7+14 = 37. In 1964 J.-R. Chen proved that any natural number can be written as the sum of 37 fifth powers of nonnegative integers.
For k = 2,3,4,... define s(k) as the smallest positive integer s such that {a(1)*x(1)^k+...+a(s)*x(s)^k: x(1),...,x(s) = 0,1,2,...} = {0,1,2,...} for some positive integers a(1), ..., a(s), and t(k) as the least positive integer t such that {a(1)*x(1)^k+...+a(t)*x(t)^k: x(1),...,x(t) = 0,1,2,...} = {0,1,2,...} for some positive integers a(1), ..., a(t) with a(1)+...+a(t) = g(k), where g(.) is given by A002804. Then s(k) <= t(k) <= g(k). Part (iii) of the conjecture in A271099 implies that t(k) <= 2k-1 for k > 2. It is easy to see that s(2) = t(2) = 4. Our computation suggests that s(3) = t(3) = 5, s(4) = t(4) = 7, s(5) = t(5) = 8 (which is smaller than 2*5-1), and s(6) = t(6) = 10. We conjecture that s(k) = t(k) for any integer k > 1, and that each natural number can be written as x(1)^6+x(2)^6+x(3)^6+2*x(4)^6+3*x(5)^6+5*x(6)^6+6*x(7)^6+10*x(8)^6+18*x(9)^6+26*x(10)^6, where x(1),x(2),...,x(10) are nonnegative integers. Note that 1+1+1+2+3+5+6+10+18+26 = 73 = g(6).
We also conjecture that any natural number can be written as s^5+t^5+2*u^5+3*v^5+4*w^5+6*x^5+8*y^5+12*z^5, with s,t,u,v,w,x,y,z nonnegative integers. Note that 1+1+2+3+4+6+8+12 = 37 = g(5). - Zhi-Wei Sun, Apr 04 2016
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REFERENCES
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J.-R. Chen, Waring's Problem for g(5)=37, Sci. Sinica 13(1964), 1547-1568.
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LINKS
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EXAMPLE
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a(1) = 1 since 1 = 0^5 + 1^5 + 2*0^5 + 3*0^5 + 4*0^5 + 5*0^5 + 7*0^5 + 14*0^5.
a(2602) = 1 since 2602 = 0^5 + 1^5 + 2*4^5 + 3*2^5 + 4*1^5 + 5*1^5 + 7*0^5 + 14*2^5.
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MATHEMATICA
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FQ[n_]:=FQ[n]=IntegerQ[n^(1/5)]
Do[r=0; Do[If[FQ[n-14z^5-7y^5-5x^5-4w^5-3v^5-2u^5-s^5], r=r+1], {z, 0, (n/14)^(1/5)}, {y, 0, ((n-14z^5)/7)^(1/5)}, {x, 0, ((n-14z^5-7y^5)/5)^(1/5)}, {w, 0, ((n-14z^5-7y^5-5x^5)/4)^(1/5)}, {v, 0, ((n-14z^5-7y^5-5x^5-4w^5)/3)^(1/5)}, {u, 0, ((n-14z^5-7y^5-5x^5-4w^5-3v^5)/2)^(1/5)}, {s, 0, ((n-14z^5-7y^5-5x^5-4w^5-3v^5-2u^5)/2)^(1/5)}]; Print[n, " ", r]; Label[aa]; Continue, {n, 0, 80}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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