cp's OEIS Frontend

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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A222949 Numbers that are a sum of four nonzero squares where the summands have no common square factor > 1.

Original entry on oeis.org

4, 7, 10, 12, 13, 15, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 30, 31, 33, 34, 35, 36, 37, 38, 39, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 97, 98, 99, 100, 101, 102
Offset: 1

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Author

Wolfdieter Lang, Mar 25 2013

Keywords

Comments

A representation of a(n) as a sum of four nonzero squares is denoted by [s(1),s(2),s(3),s(4)] with nondecreasing entries > 1 and Sum_{j=1..4} s(j)^2 = a(n). It is called primitive if gcd(s(1),s(2),s(3),s(4)) = 1. a(n) is the number with at least one such primitive representation for n, and the multiplicity m is given by the non-vanishing entries of A097203, that is A097203(a(n)).

Examples

			a(1) = 4 because 4 has the primitive representation [1, 1, 1, 1].
a(16) = 28, because 28 has the primitive representations [1, 1, 1, 5] and [1, 3, 3, 3] ([2, 2, 2, 4] is not primitive.).
4: [1, 1, 1, 1], 7: [1, 1, 1, 2], 10: [1, 1, 2, 2], 12: [1, 1, 1, 3], 13: [1, 2, 2, 2], 15: [1, 1, 2, 3], 18: [1, 2, 2, 3],
  19: [1, 1, 1, 4], 20: [1, 1, 3, 3], 21: [2, 2, 2, 3], 22: [1, 1, 2, 4], 23: [1, 2, 3, 3], 25: [1, 2, 2, 4],  26: [2, 2, 3, 3], 27: [1, 1, 3, 4], 28: [1, 1, 1, 5], [1, 3, 3, 3], ...

Crossrefs

Cf. A097203.

Formula

A097203(a(n)) is not 0.

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

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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}.

A223726 Multiplicities for A004433: sum of four distinct nonzero squares.

Original entry on oeis.org

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

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Author

Wolfdieter Lang, Mar 26 2013

Keywords

Comments

The number A004433(n) can be partitioned into four distinct parts, each of which is a nonzero square, and a(n) gives the multiplicity which is the number of different partitions of this type.

Examples

			a(1) = 1 because  A004433(1) = 30 has only one representation as sum of four distinct nonzero squares, given by the quadruple [1,2,3,4]: 1^2 + 2^2 + 3^2 + 4^2 = 30.
a(16) = 3 because for A004433(3) = 78 the three different quadruples are [1, 2, 3, 8], [1, 4, 5, 6] and [2, 3, 4, 7].
a(48) = 5 because A004433(48) = 126 has five different  representations given by the five quadruples [1, 3, 4, 10], [1, 5, 6, 8], [2, 3, 7, 8], [2, 4, 5, 9], [4, 5, 6, 7].

Crossrefs

Cf. A004433, A025428, A000414, A097203, A222949.

Formula

a(n) = k if there are k different quadruples [s(1),s(2),2(3),s(4)] with increasing positive entries with sum(s(j)^2,j=1..4) = A004433(n), n >= 1.
Showing 1-3 of 3 results.

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