A223727 -id:A223727 - cp's OEIS frontend

A223728 Multiplicities for A223727: primitive sums of four distinct nonzero squares.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 3, 2, 1, 1, 1, 3, 2, 1, 3, 1, 1, 1, 2, 1, 1, 2, 1, 5, 1, 2, 3, 1, 1, 2, 3, 1, 2, 1, 1, 2, 4, 2, 1, 2, 3, 1, 5, 2, 2, 2, 2, 2, 4, 3, 1, 4, 1, 1, 4, 2, 2, 2, 5, 3, 1, 6, 3, 3, 1, 2, 1, 1, 4, 4, 2, 5, 1, 3, 7, 3, 2

Offset: 1

Wolfdieter Lang, Mar 27 2013

nonn

A223728(n) has a(n) different primitive representations as sum of four distinct nonzero squares, n>=1.

			a(16) = 3 because A223727(16) = 78 has three s-quadruples, namely [1, 2, 3, 8], [1, 4, 5, 6] and [2, 3, 4, 7].
a(23) = 2 from A223727(23) =  90 with s-quadruples [1, 2, 6, 7] and [1, 3, 4, 8].

Cf. A223727, A223726.

a(n) = k if there are k different solutions for A223728(n) = sum(s(j)^2, j=1..4), with 0 < s(1) < s(2) < s(3) < s(4) and gcd(s(1),s(2),s(3),s(4)) = 1.

A259058 Numbers that are representable in at least two ways as sums of four distinct nonvanishing squares.

Original entry on oeis.org

454, 530, 614, 706, 806, 914, 1030, 1154, 1286, 1426, 1574, 1730, 1894, 2066, 2246, 2434, 2630, 2834, 3046, 3266, 3494, 3730, 3974, 4226, 4486, 4754, 5030, 5314, 5606, 5906, 6214, 6530, 6854, 7186, 7526, 7874, 8230, 8594, 8966, 9346, 9734

Offset: 0

Wolfdieter Lang, Aug 12 2015

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.
  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.
There may be other numbers having this property.
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

			n=0: 454 = 1^2 + 8^2 + 10^2 + 17^2 = 2^2 + 5^2 + 13^2 + 16^2.
n=2: 614 = 3^2 + 10^2 + 12^2 + 19^2 = 4^2 + 7^2 + 15^2 + 18^2.

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

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

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

[4*n^2 + 72*n + 454: n in [0..50]]; // Vincenzo Librandi, Aug 13 2015

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

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

a(n)=4*n^2+72*n+454 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 4*n^2 + 72*n + 454 = 2*A259059(n). See the comment for the sum of four squares in two ways.
O.g.f.: 2*(227 - 416*x + 193*x^2)/(1-x)^3.
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). - Vincenzo Librandi, Aug 13 2015

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A223728 Multiplicities for A223727: primitive sums of four distinct nonzero squares.

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A259058 Numbers that are representable in at least two ways as sums of four distinct nonvanishing squares.

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