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A216280
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Number of nonnegative solutions to the equation x^4 + y^4 = n.
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2
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1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
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OFFSET
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1,635318657
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COMMENTS
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The first n with a(n) > 1 is 635318657 = 41 * 113 * 241 * 569, with a(635318657) = 2. Izadi, Khoshnam, & Nabardi show that for any n with a(n) > 1, the elliptic curve y^2 = x^3 - nx has rank at least 3. According to gp, y^2 = x^3 - 635318657x has analytic rank 4 (and first nonzero derivative around 35741.7839). - Charles R Greathouse IV, Jan 12 2017
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LINKS
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MATHEMATICA
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Reap[For[n = 1, n <= 1000, n++, r = Reduce[0 <= x <= y && x^4 + y^4 == n, {x, y}, Integers]; sols = Which[r === False, 0, r[[0]] == And, 1, r[[0]] == Or, Length[r], True, Print[n, " ", r]]; If[sols != 0, Print[n, " ", sols, " ", r]]; Sow[sols]]][[2, 1]] (* Jean-François Alcover, Feb 22 2019 *)
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PROG
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(PARI) a(n)=my(t=thue(thueinit('x^4+1, 1), n)); sum(i=1, #t, t[i][1]>=0 && t[i][2]>=t[i][1]) \\ Charles R Greathouse IV, Jan 12 2017
(PARI) first(n)=my(T=thueinit('x^4+1, 1), v=vector(n), t); for(k=1, n, t=thue(T, k); v[k]=sum(i=1, #t, t[i][1]>=0 && t[i][2]>=t[i][1])); v \\ Charles R Greathouse IV, Jan 12 2017
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CROSSREFS
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Cf. A004831 (positions of nonzero terms).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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