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%I #28 Dec 10 2023 09:19:21
%S 1,11,101,13361,1169341,1612186411,1624763543401,20188985439712961,
%T 240020196429554642201,29891946989942513908518251,
%U 3506790234728288196345900732301,5190947078637547438603476743093680561
%N Positive integers k such that k^2 = (m^5 + n^5)/(m + n) for some coprime integers m, n.
%C This sequence probably contains no more than 5 primes.
%H Kevin Acres and David Broadhurst, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;5b5c3a3a.1102">Rational points on y^2 = x^3 + 10*x^2 + 5*x</a>
%H Dave Rusin, <a href="http://www.math.niu.edu/~rusin/known-math/96/two_fifths">Re: Help with diophantine equ.</a>, sci.math newsgroup [Broken link]
%H Dave Rusin, <a href="/A156668/a156668.txt">Re: Help with diophantine equ.</a>, sci.math newsgroup [Cached copy]
%F Numerators of rational numbers (81*x^4 + 540*x^3 - 8370*x^2 + 33900*x - 47975)/(9*x^2 - 150*x + 445)^2, where x ranges over abscissas of rational points on the elliptic curve y^2 = x^3 - 85/3*x + 1550/27.
%e 13361 belongs to this sequence since 13361^2 = (35^5 + 123^5) / (35 + 123) with gcd(35, 123)=1.
%o (PARI) { a(k) = local(P=ellpow(ellinit([0,10,0,5,0]),[-1,2],k),s,t); s=P[1]^2;t=abs(numerator(P[2]^4/s-80*s)); while(t%2==0,t=t/2); t } /* _David Broadhurst_ */
%Y Cf. A156669, A156670, A186082, A339325 (m), A339326 (n).
%K nonn
%O 1,2
%A _Max Alekseyev_, Feb 13 2009
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