

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
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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*y3)/2 <= i <= (5*y+3)/2} i^2 = Sum_{x2 <= 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+m1];
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 (* JeanFrançois 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



