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A186082 Numbers k such that log(A156668(k)*(1 + k mod 2))/k^2 is smaller than for any prior k.

%I A186082 #25 Sep 03 2025 20:09:58
%S A186082 1,2,3,5,10,13,18,31,49,98,116,232,281,397,678,1075,2150,3225,4300,
%T A186082 5375,5772,6847,7922,8997,17994,19069
%N A186082 Numbers k such that log(A156668(k)*(1 + k mod 2))/k^2 is smaller than for any prior k.
%H A186082 David Broadhurst, <a href="/A186082/b186082.txt">Table of n for n = 1..1000</a>
%H A186082 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>
%e A186082 [1,  0.6931471805599453094172321215]
%e A186082 [2,  0.5994738181995926360154858945]
%e A186082 [3,  0.5898075219334671955890478209]
%e A186082 [5,  0.5866039232314788114510488867]
%e A186082 [10, 0.5865963134453746145925657102]
%e A186082 [13, 0.5864356249925781873672553233]
%e A186082 [18, 0.5864232678262123920533476064]
%o A186082 (PARI)
%o A186082 T(k)={
%o A186082     local(P=ellpow(ellinit([0,10,0,5,0]),[-1,2],k),s,t);
%o A186082     s=P[1]^2;
%o A186082     t=abs(numerator(P[2]^4/s-80*s));
%o A186082     while(t%2==0,t=t/2;);
%o A186082     t;
%o A186082 }
%o A186082 M(k)=1+k%2;
%o A186082 V(k)=log(M(k)*T(k))/k^2;
%o A186082 {
%o A186082     lowest=1;
%o A186082     for(i=1,20000,
%o A186082         l=V(i);
%o A186082         if(l<lowest,
%o A186082             lowest=l;
%o A186082             print([i,l]);
%o A186082         );
%o A186082     );
%o A186082 }
%Y A186082 Cf. A156668.
%K A186082 nonn,changed
%O A186082 1,2
%A A186082 _Kevin T. Acres_, Feb 12 2011