A077119 a(n) = A077118(n) - n^3.

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

Offset: 0

Author

Reinhard Zumkeller, Oct 29 2002

Keywords

• sign

Comments

a(n)=0 iff n = m^(6*k).

Values d=x^3-y^2 of extremal points of elliptic Mordell curves. Definition for extremal points see A200656. Each value x has only one value of distance d when coordinate x is extremal point, but for many fixed distances d, the elliptic curve has more than 1 extremal point. - Artur Jasinski, 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 Jasinski, Dec 08 2011

Examples

			A077118(10)=1024=32^2 is the nearest square to 10^3=1000, therefore a(10)=1024-1000=24.
		

Crossrefs

Cf. A000578, A077118, A077111.

|a(n)| = A002938(n).

Programs

Magma

[Round(Sqrt(n^3))^2-n^3: n in [0..60]]; // Vincenzo Librandi, Mar 24 2015

Maple

A077119 := proc(n)
    (round( sqrt(n^3) ))^2-n^3 ;
end proc: # R. J. Mathar, Jan 18 2021

Mathematica

Table[Round[Sqrt[x^3]]^2 - x^3, {x, 0, 100}]  (* Artur Jasinski, Nov 30 2011 *)

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

Formula

a(n) = if A077116(n) < A070929(n) then -A077116(n) else A070929(n).