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%I #27 Sep 08 2022 08:45:07
%S 0,0,1,-2,0,-4,9,18,17,0,24,-35,36,12,-40,-11,0,-13,-56,30,-79,-45,
%T -39,-67,100,0,113,-83,-48,-53,-104,138,-7,163,-100,-26,0,-28,-116,
%U 217,9,248,-104,17,80,79,8,-139,297,0,316,-155,17,119,145,89,-55
%N a(n) = A077118(n) - n^3.
%C a(n)=0 iff n = m^(6*k).
%C Values d=x^3-y^2 of extremal points of elliptic Mordell curves. Definition extremal points see A200656. Each value x have only one value of distance d when coordinate x is extremal point, but for many fixed distances d elliptic curve have more than 1 extremal point. - _Artur Jasiński_, Nov 30 2011
%C Theorem (*Artur Jasinski*): If a(n)>0 then a(n)<(4n^(3/2)-1)/4 for every n. If a(n)<0 then a(n)>(-4n^(3/2)-1)/4 for every n. a(n)=0 then n is perfect square. - _Artur Jasiński_, Dec 08 2011
%F a(n) = if A077116(n) < A070929(n) then -A077116(n) else A070929(n).
%e A077118(10)=1024=32^2 is the nearest square to 10^3=1000, therefore a(10)=1024-1000=24.
%p A077119 := proc(n)
%p (round( sqrt(n^3) ))^2-n^3 ;
%p end proc: # _R. J. Mathar_, Jan 18 2021
%t Table[Round[Sqrt[x^3]]^2 - x^3, {x, 0, 100}] (* _Artur Jasinski_, Nov 30 2011 *)
%o (Magma) [Round(Sqrt(n^3))^2-n^3: n in [0..60]]; // _Vincenzo Librandi_, Mar 24 2015
%o (Python)
%o from math import isqrt
%o def A077119(n): return ((m:=isqrt(k:=n**3))+int((k-m*(m+1)<<2)>=1))**2-k # _Chai Wah Wu_, Jul 29 2022
%Y Cf. A000578, A077118, A077111.
%Y |a(n)| = A002938(n).
%K sign
%O 0,4
%A _Reinhard Zumkeller_, Oct 29 2002
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