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0, 0, 1, -2, 0, -4, 9, 18, 17, 0, 24, -35, 36, 12, -40, -11, 0, -13, -56, 30, -79, -45, -39, -67, 100, 0, 113, -83, -48, -53, -104, 138, -7, 163, -100, -26, 0, -28, -116, 217, 9, 248, -104, 17, 80, 79, 8, -139, 297, 0, 316, -155, 17, 119, 145, 89, -55
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OFFSET
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0,4
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COMMENTS
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a(n)=0 iff n = m^(6*k).
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
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
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LINKS
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FORMULA
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EXAMPLE
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A077118(10)=1024=32^2 is the nearest square to 10^3=1000, therefore a(10)=1024-1000=24.
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MAPLE
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(round( sqrt(n^3) ))^2-n^3 ;
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MATHEMATICA
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Table[Round[Sqrt[x^3]]^2 - x^3, {x, 0, 100}] (* Artur Jasinski, Nov 30 2011 *)
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PROG
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(Python)
from math import isqrt
def A077119(n): return ((m:=isqrt(k:=n**3))+int((k-m*(m+1)<<2)>=1))**2-k # Chai Wah Wu, Jul 29 2022
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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