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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A259059 One half of numbers representable in at least two different ways as sums of four distinct nonvanishing squares. See A259058 for these numbers and their representations.

Original entry on oeis.org

227, 265, 307, 353, 403, 457, 515, 577, 643, 713, 787, 865, 947, 1033, 1123, 1217, 1315, 1417, 1523, 1633, 1747, 1865, 1987, 2113, 2243, 2377, 2515, 2657, 2803, 2953, 3107, 3265, 3427, 3593, 3763, 3937, 4115, 4297, 4483, 4673, 4867, 5065, 5267, 5473
Offset: 0

Views

Author

Wolfdieter Lang, Aug 12 2015

Keywords

Comments

There may be other numbers with this property.

References

  • W. Sierpiński, 250 Problems in Elementary Number Theory, American Elsevier Publ. Comp., New York, PWN-Polish Scientific Publishers, Warszawa, 1970, Problem 227, p. 20 and p. 110.

Links

Crossrefs

Cf. A259058.

Programs

Formula

a(n) = A259058(n)/2.
a(n) = 2*n^2 + 36*n + 227.
O.g.f.: (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

A223727 Numbers which are a sum of four distinct nonzero squares where the summands have no common factor > 1.

Original entry on oeis.org

30, 39, 46, 50, 51, 54, 57, 62, 63, 65, 66, 70, 71, 74, 75, 78, 79, 81, 84, 85, 86, 87, 90, 91, 93, 94, 95, 98, 99, 102, 105, 106, 107, 109, 110, 111, 113, 114, 116, 117, 118, 119, 121, 122, 123, 125, 126, 127, 129, 130, 131, 133, 134, 135, 137, 138, 139, 140
Offset: 1

Views

Author

Wolfdieter Lang, Mar 27 2013

Keywords

Comments

A primitive representation of a number m as a sum of four distinct nonzero squares is determined from a quadruple [s(1), s(2), s(3), s(4)] of integers with 0 < s(1) < s(2) < s(3) < s(4) with gcd(s(1),s(2),s(3),s(4)) = 1, and m = sum(s(j)^2, j=1..4). If m has such a primitive representation then k^2*m, with integer k > 0, has trivially a non-primitive representation. Therefore primitive representations are of interest.
For the multiplicities see A223728.
This sequence is a proper subset of A004433. The first entry of A004433 missing here is 120 = A004433(43). The first common entry with different multiplicity is A004433(72) = 156 = a(71) with two primitive representations with quadruples
[1, 3, 5, 11] and [1, 5, 7, 9]. [2, 4, 6, 10] = 2*[1, 2, 3, 5]is a non-primitive representation due to 156 = 4*39.

Examples

			a(1) = 30 because the numbers 0,...,29 have no representation as a sum of four distinct nonzero squares, and 30 has one representation given by the quadruple [1,2,3,4] which is primitive.
a(16) = 78 has three such representations given by the quadruples  [1, 2, 3, 8], [1, 4, 5, 6] and [2, 3, 4, 7] which are all primitive. Hence A223728(16) = 3. This is the first entry with more than one (primitive) representation.
a(23) = 90 has multiplicity 2 = A223728 because there are two primitive quadruples [1, 2, 6, 7] and [1, 3, 4, 8].
a(71) = 156 has multiplicity A223728(71) = 2 (see a comment above).

Crossrefs

Cf. A222949, A097203, A223728, A259058 (multiplicity >= 2 instances).

Formula

This sequence are the increasingly ordered members of the set {m an integer | m = 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}.

A259060 Numbers that are representable in at least two ways as sums of four distinct nonvanishing cubes.

Original entry on oeis.org

6426, 7900, 9614, 11592, 13858, 16436, 19350, 22624, 26282, 30348, 34846, 39800, 45234, 51172, 57638, 64656, 72250, 80444, 89262, 98728, 108866, 119700, 131254, 143552, 156618, 170476, 185150, 200664, 217042, 234308, 252486, 271600, 291674
Offset: 0

Views

Author

Wolfdieter Lang, Aug 12 2015

Keywords

Comments

This is the second part of Exercise 229 in Sierpiński's problem book. See p. 20, and p. 110 for the solution. He uses the identity (n-8)^3 + (n-1)^3 + (n+1)^3 + (n+8)^3 = 4*n^3 + 390 = (n-7)^3 + (n-4)^3 + (n+4)^3 + (n+7)^3, for n >= 9.
Here n is replaced by n + 9: (n+1)^3 + (n+8)^3 + (n+10)^3 + (n+17)^3 = 4*n^3 + 108*n^2 + 1362*n + 6426 = (n+2)^3 + (n+5)^3 + (n+13)^3 + (n+16)^3, for n >= 0.
There may be other numbers with this properties.
Because the summands have no common factor > 1 each of these two representations is called primitive. - Wolfdieter Lang, Aug 20 2015

Examples

			a(0) = 6426 = 1^3 + 8^3 + 10^3 + 17^3 = 2^3 + 5^3 + 13^3 + 16^3.
a(1) = 7900 = 2^3 + 9^3 + 11^3 + 18^3 = 3^3 + 6^3 + 14^3 + 17^3.

References

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

Links

Crossrefs

Cf. A261241, A259058 (squares).

Programs

  • Magma
    [(2*(n+9))*(2*n^2+36*n+357): n in [0..50]]; // Vincenzo Librandi, Aug 13 2015
  • Magma
    I:=[6426,7900,9614,11592]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Aug 13 2015
  • Mathematica
    CoefficientList[Series[2 (3213 - 8902 x + 8285 x^2 - 2584 x^3)/(1 - x)^4, {x, 0, 50}], x] (* Vincenzo Librandi, Aug 13 2015 *)
LinearRecurrence[{4,-6,4,-1},{6426,7900,9614,11592},40] (* Harvey P. Dale, Sep 30 2016 *)

Formula

a(n) = (2*(n+9))*(2*n^2+36*n+357) = 2*A261241(n), n >= 0. See the comment for the sum of four distinct cubes in two different ways.
O.g.f.: 2*(3213 - 8902*x + 8285*x^2 - 2584*x^3) / (1-x)^4.
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Vincenzo Librandi, Aug 13 2015
Showing 1-3 of 3 results.

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