A307937 Numbers that can be written as the sum of four or more consecutive squares in more than one way.

3655, 3740, 4510, 4760, 5244, 5434, 5915, 7230, 7574, 8415, 11055, 11900, 12524, 14905, 17484, 18879, 19005, 19855, 20449, 20510, 21790, 22806, 23681, 25580, 25585, 27230, 27420, 28985, 31395, 34224, 37114, 39606, 41685, 42419, 44919, 45435, 45955, 48026, 48139, 48225, 49015, 53941, 57164, 62006

Offset

1,1

Comments

Numbers that are in A174071 in two or more ways.

The first number with more than two representations as a sum of four or more consecutive positive squares is 147441 = 18^2 + ... + 76^2 = 29^2 + ... + 77^2 = 85^2 + ... + 101^2.

If x = 2*A049629(n) and y = A007805(n) for n >= 1 (satisfying the Pell equation x^2 - 5*y^2 = -1), then the sequence contains 5*x^2+10 = Sum_{(5*y-3)/2 <= i <= (5*y+3)/2} i^2 = Sum_{x-2 <= i <= x+2} i^2 = 25*y^2 + 5.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

a(1) = 3655 is in the sequence because 3655 = 8^2 + ... + 22^2 = 25^2 + ... + 29^2.

Maple

N:= 10^5: # to get all terms <= N
R:= 'R':
dups:= NULL:
for m from 4 while m*(m+1)*(2*m+1)/6 <= N do
   for k from 1 do
       v:= m*(6*k^2 + 6*k*m + 2*m^2 - 6*k - 3*m + 1)/6;
       if v > N then break fi;
       if assigned(R[v]) then
         dups:= dups, v;
       else
         R[v]:= [k, k+m-1];
       fi;
od od:
sort(convert({dups}, list));

Mathematica

M = 10^5;
dups = {}; Clear[rQ]; rQ[_] = False;
For[m = 4, m(m+1)(2m+1)/6 <= M, m++, For[k = 1, True, k++, v = m(6k^2 + 6k m + 2m^2 - 6k - 3m + 1)/6; If[v > M, Break[]]; If[rQ[v], AppendTo[dups, v], rQ[v] = True]]];
dups // Sort (* Jean-François Alcover, May 07 2019, after Robert Israel *)

Crossrefs

Cf. A007805, A049629, A174071.

Sequence in context: A212852 A183781 A252678 * A190923 A255088 A283732

Adjacent sequences: A307934 A307935 A307936 * A307938 A307939 A307940

Keyword

nonn

Author

Robert Israel, May 06 2019

Status

approved