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Numbers that are representable in at least two ways as sums of four distinct nonvanishing squares.
3

%I #28 Sep 08 2022 08:46:13

%S 454,530,614,706,806,914,1030,1154,1286,1426,1574,1730,1894,2066,2246,

%T 2434,2630,2834,3046,3266,3494,3730,3974,4226,4486,4754,5030,5314,

%U 5606,5906,6214,6530,6854,7186,7526,7874,8230,8594,8966,9346,9734

%N Numbers that are representable in at least two ways as sums of four distinct nonvanishing squares.

%C This is part one of Exercise 229 in Sierpiński's problem book. See p. 20 and p. 110 for the solution. He uses the identity (n-8)^2 + (n-1)^2 + (n+1)^2 + (n+8)^2 = 4*n^2 + 130 = (n-7)^2 + (n-4)^2 + (n+4)^2 + (n+7)^2, for n >= 9.

%C Here n was replaced by n + 9: (n+1)^2 + (n+8)^2 +(n+10)^2 + (n+17)^2 = 4*n^2 + 72*n + 454 = (n+2)^2 + (n+5)^2 + (n+13)^2 + (n+16)^2, for n >= 0.

%C There may be other numbers having this property.

%C Because the summands have no common factor > 1 each of these two representations is called primitive. Therefore, this is a proper subsequence of A223727, hence of A004433. - _Wolfdieter Lang_, Aug 20 2015

%D W. Sierpiński, 250 Problems in Elementary Number Theory, American Elsevier Publ. Comp., New York, PWN-Polish Scientific Publishers, Warszawa, 1970.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = 4*n^2 + 72*n + 454 = 2*A259059(n). See the comment for the sum of four squares in two ways.

%F O.g.f.: 2*(227 - 416*x + 193*x^2)/(1-x)^3.

%F a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). - _Vincenzo Librandi_, Aug 13 2015

%e n=0: 454 = 1^2 + 8^2 + 10^2 + 17^2 = 2^2 + 5^2 + 13^2 + 16^2.

%e n=2: 614 = 3^2 + 10^2 + 12^2 + 19^2 = 4^2 + 7^2 + 15^2 + 18^2.

%t CoefficientList[Series[2 (227 - 416 x + 193 x^2)/(1 - x)^3, {x, 0, 50}], x] (* _Vincenzo Librandi_, Aug 13 2015 *)

%o (Magma) [4*n^2 + 72*n + 454: n in [0..50]] /* or */ I:=[454, 530, 614]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]]; // _Vincenzo Librandi_, Aug 13 2015

%o (PARI) a(n)=4*n^2+72*n+454 \\ _Charles R Greathouse IV_, Jun 17 2017

%Y Cf. A259059, A223727, A004433, A259060 (four cubes).

%K nonn,easy

%O 0,1

%A _Wolfdieter Lang_, Aug 12 2015