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A307937 Numbers that can be written as the sum of four or more consecutive squares in more than one way. 2
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified August 11 18:37 EDT 2020. Contains 336428 sequences. (Running on oeis4.)