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A190581 Value of z in the Diophantine equation x^3 + y^3 = n*z^3 (with x>0 and minimal and x >= y and y != 0).

%I A190581 #26 Aug 05 2025 10:02:38
%S A190581 1,21,1,1,39,3,294,7,1,7,9954,1,1,57,42,582,182,1,1,129,2,3,6111,
%T A190581 197028,217,7083,1,3,1,1,1323,620505,3318,13,43,3606,1302,3,111,
%U A190581 330498,3,216266610,13,273,1,5733,590736058375050,3,1,117,1014,25767,19,37,1878,1029364,1,37045412880,1,1,1,11285694
%N A190581 Value of z in the Diophantine equation x^3 + y^3 = n*z^3 (with x>0 and minimal and x >= y and y != 0).
%C A190581 A190356(n)^3 + y^3 = A020898(n)*a(n)^3. Unknown y corresponds to sequence A190580.
%C A190581 The 4 sequences A020898 [i.e. n], A190356 [i.e. x], A190580 [i.e. y] and A190581 [i.e. z] satisfy the equation A190356(n)^3 + A190580(n)^3 = A020898(n) * A190581(n)^3
%H A190581 Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/csolve/fermat.pdf">On a Generalized Fermat-Wiles Equation</a> [broken link]
%H A190581 Steven R. Finch, <a href="http://web.archive.org/web/20010602030546/http://www.mathsoft.com/asolve/fermat/fermat.html">On a Generalized Fermat-Wiles Equation</a> [From the Wayback Machine]
%H A190581 Nakao Hisayasu, <a href="https://www.kaynet.or.jp/~kay/misc/nna2.html">Rational Points on Elliptic Curves: x^3+y^3=n</a> (nna2.html up to nna22.html)
%H A190581 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/ec/eca1/x3y3s.txt">Solutions of Diophantine equation x^3+y^3=A.z^3 ...</a>
%e A190581 a(18) = 1  because  A020898(18) = 35 and 3^3 + 2^3 = 35*1^3.
%t A190581 a[n_] := z /. ToRules[ Reduce[ z > 0 && A190356[[n]]^3 + A190580[[n]]^3 == A020898[[n]]*z^3, z, Integers]]; Table[a[n] , {n, 1, 62}]
%Y A190581 Cf. A020898, A060838, A190356, A190580
%K A190581 nonn
%O A190581 1,2
%A A190581 _Jean-François Alcover_, May 13 2011