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Numbers that can be written as a sum of at least 4 consecutive positive squares.
4

%I #7 May 06 2019 20:54:49

%S 30,54,55,86,90,91,126,135,139,140,174,190,199,203,204,230,255,271,

%T 280,284,285,294,330,355,366,371,380,384,385,415,446,451,476,492,501,

%U 505,506,510,534,559,595,615,620,630,636,645,649,650,679,728,730,734,764

%N Numbers that can be written as a sum of at least 4 consecutive positive squares.

%C Numbers of the form m*(6*k^2 + 6*k*m + 2*m^2 - 6*k - 3*m + 1)/6 for some m>=4 and k>=1. - _Robert Israel_, May 06 2019

%H Robert Israel, <a href="/A174071/b174071.txt">Table of n, a(n) for n = 1..10000</a>

%e 30=1^2+2^2+3^2+4^2, 54=2^2+3^2+4^2+5^2, 55=1^2+2^2+3^2+4^2+5^2, ...

%p N:= 1000: # to get all terms <= N

%p Res:= NULL:

%p for m from 4 while m*(m+1)*(2*m+1)/6 <= N do

%p for k from 1 do

%p v:= m*(6*k^2 + 6*k*m + 2*m^2 - 6*k - 3*m + 1)/6;

%p if v > N then break fi;

%p Res:= Res, v;

%p od od:

%p sort(convert({Res},list)); _Robert Israel_, May 06 2019

%t max=60^2;lst={};Do[z=n^2+(n+1)^2+(n+2)^2;Do[z+=(n+x)^2;If[z>max,Break[]];AppendTo[lst,z],{x,3,Sqrt[max]/2}],{n,Sqrt[max]/2}];Union[lst]

%Y Cf. A111774, A138591, A174069, A174070, A307937.

%K nonn

%O 1,1

%A _Vladimir Joseph Stephan Orlovsky_, Mar 06 2010

%E Edited by _Robert Israel_, May 06 2019