A223733 -id:A223733 - cp's OEIS frontend

A223731 All positive numbers that are primitive sums of three nonzero squares.

3, 6, 9, 11, 14, 17, 18, 19, 21, 22, 26, 27, 29, 30, 33, 34, 35, 38, 41, 42, 43, 45, 46, 49, 50, 51, 53, 54, 57, 59, 61, 62, 65, 66, 67, 69, 70, 73, 74, 75, 77, 78, 81, 82, 83, 86, 89, 90, 91, 93, 94, 97, 98, 99, 101, 102, 105, 106, 107, 109, 110, 113, 114, 115, 117, 118

Offset: 1

Wolfdieter Lang, Apr 05 2013

nonn

These are the ordered numbers for which A223730 is not zero. The multiplicity for the number a(n) is A223730(a(n)).
According to the Halter-Koch reference the present sequence lists the ordered positive integers satisfying i) n not 0, 4, or 7 (mod 8) (see p.10, formula for r_3(n) attributed to A. Schinzel) and ii) n not from the set {1,2,5,10,13,25,37,58,85,130} with possibly one more positive integer member of this set which has to be >= 5*10^10 (if it exists at all). (Korollar 1. (b), p. 13). For this set see also A051952.
The first members with multiplicity 1 (precisely one representation) are 3, 6, 9, 11, 14, 17, 18, 19, 21, 22, 26, 27, 29, 30, 34, 35, 42, 43, 45, 46, 49, 50, 53, 61, 65, 67 ... A223732.
The first members with multiplicity 2 are 33, 38, 41, 51, 54, 57, 59, 62, 69, 74, 77, 81, 83, 90, 94, 98, 99, ... A223733.
The first members with multiplicity 3 are 66, 86, 89, 101, 110, 114, 131, 149, 153, 166, 171, 173, ... A223734.
For the complement see A223735.

			a(12) = 27 because 27 is the 12th number for which A223730 is nonzero. Because A223730(27) = 1  there is only one primitive sum of three nonzero squares which is 27 denoted by [1,1,5]:
  1^2 + 1^2 + 5^2 = 27.
a(28) = 54 has two primitive representations in question, namely [1,2,7] and [2,5,5]. A223730(54) = 2. The representation [3,3,6] is not primitive because gcd(3,3,6) = 3 not 1.
a(34) = 66 has three representations in question, namely [1,1,8], [1,4,7] and [4,5,5].

Alois P. Heinz, Table of n, a(n) for n = 1..1000
F. Halter-Koch, Darstellung natürlicher Zahlen als Summe von Quadraten, Acta Arith. 42 (1982) 11-20.

Cf. A223730, A000408 (non-primitive case), A223735 (complement).

threeSquaresQ[n_] := Select[ PowersRepresentations[n, 3, 2], Times @@ #1 != 0 && GCD @@ #1 == 1 & ] != {}; Select[Range[120], threeSquaresQ] (* Jean-François Alcover, Jun 21 2013 *)

The sequence a(n) is obtained from the ordered set
  {m positive integer | m = a^2 + b^2 + c^2 , a,b,c integer, 0 < a <= b <= c, gcd(a,b,c) = 1} with entries appearing only once.
Conjectured g.f.: (x^77 +2*x^76 -2*x^75 +x^74 -x^73 -x^72 +2*x^50 -x^49 +2*x^47 -2*x^46 -x^45 +x^34 +2*x^33 -2*x^32 +x^31 -x^30 -x^29 +2*x^22 -x^21 +2*x^19 -2*x^18 -x^17 +3*x^15 -2*x^14 +x^13 -x^12 -x^10 +2*x^9 +2*x^7 +2*x^6 -3*x^4 -2*x^3 -3*x^2 -3*x -3)*x / (-x^6 +x^5 +x -1). - Alois P. Heinz, Apr 06 2013

A223730 Multiplicities for representations of positive numbers n as primitive sums of three nonzero squares.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 2, 1, 1, 0, 0, 2, 0, 0, 2, 1, 1, 0, 1, 1, 0, 0, 1, 1, 2, 0, 1, 2, 0, 0, 2, 0, 2, 0, 1, 2, 0, 0, 1, 3, 1, 0, 2, 1, 0, 0, 1, 2, 1, 0, 2, 1, 0, 0, 2, 1, 2, 0, 0, 3, 0, 0, 3, 2, 1, 0, 1, 2, 0, 0, 1, 2, 2, 0, 3, 2, 0, 0, 2, 1, 2, 0, 1, 3, 0, 0, 2, 3, 1, 0, 2, 2, 0, 0, 2, 2, 2, 0, 2, 2, 0, 0, 4, 0, 3, 0, 1, 4

Offset: 1

Wolfdieter Lang, Apr 04 2013

nonn

Primitive sums of three nonzero squares a^2 + b^2 + c^2, with positive integers a, b and c, satisfy gcd(a,b,c) = 1. (coprimality of the three squares).
a(n) gives the number of different representations (multiplicities) of the number n >= 1 as primitive sums of three nonzero squares. If a(n) = 0 there is no such representation for n. The numbers n with a(n) not vanishing are given in A223731. The ones with a(n) = 1, 2 and 3 are in A223732, A223733 and A223734, respectively.
For the multiplicities of the positive numbers as sums of three nonzero squares see A025427. The numbers with A025427(n) >= 1 are given in A000408.
A corollary in the Halter-Koch reference (Korollar 1. (b) on p. 13) states for the positive numbers n, not 0, 4, 7 (mod 8) [otherwise n cannot be a primitive sum of three nonzero squares; see p. 11, the r_3(n) formula]:  n is not the sum of three positive coprime squares if and only if n is from the set T := {1, 2, 5, 10, 13, 25, 37, 58, 85, 130, ?}, with ? possibly a number >= 5*10^10 . Therefore a(n) = 0 if and only if n >= 1 is of the form mentioned in this corollary: i)  0, 4, 7 (mod 8) or ii) in the set T.
For representations of n as a sum of three nonzero squares see the Grosswald reference, Theorem 7, p. 79. There also the above mentioned set T appears and for the Conjecture it is assumed that the extra eleventh member of T is absent.

			a(12) = 0 because the only representation of 12 as a sum of three nonzero squares is given by [2,2,2], i.e., 12 = 2^2 + 2^2 + 2^2, but this is not a primitive sum because gcd(2,2,2) = 2, not 1. Such a situation appears for n = 12, 24, 36, 44, 48, 56, 68, 72, 76, 84, 88, 96, ... For these numbers A025427(n) = 1 and a(n) = 0.
a(27) = 1 because the only primitive representation of 27 as a sum of three nonzero squares is denoted by [1,1,5]. The representation [3,3,3] is not primitive.

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
F. Halter-Koch, Darstellung natürlicher Zahlen als Summe von Quadraten, Acta Arith. 42 (1982) 11-20.

Cf. A223731, A025427 (non-primitive case), A223732, A223733, A223734.

with(numtheory):
b:= proc(n, i, t, s) option remember;
      `if`(n=0, `if`(t=0 and s={}, 1, 0), `if`(i=1, `if`(t=n, 1, 0),
      `if`(t*i^2xn, 0, b(n-i^2, i, t-1, `if`(s={1}, factorset(i),
       s intersect factorset(i)))))))
    end:
a:= n-> b(n, isqrt(n), 3, {1}):
seq(a(n), n=1..200);  # Alois P. Heinz, Apr 06 2013

a[n_] := Select[ PowersRepresentations[n, 3, 2], Times @@ # != 0 && GCD @@ # == 1 &] // Length; Table[a[n], {n, 1, 134}] (* Jean-François Alcover, Jun 21 2013 *)

a(n) = 0 if there is no representation of n as a primitive sum of three nonzero squares. a(n) = k >= 1 if there are k distinct such representations for n.

A223734 Positive numbers that are the sum of three nonzero squares with no common factor > 1 in exactly three ways.

Original entry on oeis.org

66, 86, 89, 101, 110, 114, 131, 149, 153, 166, 171, 173, 174, 179, 182, 185, 186, 189, 198, 219, 221, 222, 227, 233, 234, 237, 241, 242, 245, 258, 261, 270, 274, 286, 291, 294, 302, 305, 309, 318, 323, 334, 338, 347, 349, 361, 363, 366, 377, 378, 387, 405, 410

Offset: 1

Wolfdieter Lang, Apr 05 2013

nonn

These are the increasingly ordered numbers a(n) for which A223730(a(n)) = 3. See also A223731. These are the numbers n with exactly three representation as a primitive sum of three nonzero squares (not taking into account the order of the three terms, and the number to be squared for each term is taken positive).

Conjecture: a(185) = 4075 = 31^2 + 33^2 + 45^2 = 23^2 + 39^2 + 45^2 = 5^2 + 9^2 + 63^2 is the largest element of this sequence. - Alois P. Heinz, Apr 06 2013

			a(1) = 66 because the smallest number n with A223730(n) = 3 is 66. The three solutions for m = 66 are denoted by [1,1,8], [1,4,7] and [4,5,5].
For n=2..10 the three representations of a(n) are given by
n=2,   86:  [1, 2, 9],  [1, 6, 7], [5, 5, 6],
n=3,   89:  [2, 2, 9],  [2, 6, 7], [3, 4, 8],
n=4,  101:  [1, 6, 8],  [2, 4, 9], [4, 6, 7],
n=5,  110:  [1, 3, 10], [2, 5, 9], [5, 6, 7],
n=6,  114:  [1, 7, 8],  [4, 7, 7], [5, 5, 8],
n=7,  131:  [1, 3, 11], [1, 7, 9], [5, 5, 9],
n=8,  149:  [1, 2, 12], [2, 8, 9], [6, 7, 8],
n=9,  153:  [2, 7, 10], [4, 4, 11], [5, 8, 8],
n=10: 166:  [2, 9, 9],  [3, 6, 11], [6, 7, 9].
For n = 153 there is also the non-primitive representation [6,6,9] = 3*[2,2,3] not counted here.

Alois P. Heinz, Table of n, a(n) for n = 1..185

Cf. A223730, A223731, A223732, A223733.

threeSquaresCount[n_] := Length[ Select[ PowersRepresentations[n, 3, 2], Times @@ #1 != 0 && GCD @@ #1 == 1 & ]]; Select[ Range[500], threeSquaresCount[#] == 3 &] (* Jean-François Alcover, Jun 21 2013 *)

This sequence lists the increasingly ordered distinct members of the set S3:= {m positive integer | m = a^2 + b^2 + c^2, 0 < a <= b <= c, and there are exactly three different solutions for this m}.

A223735 Positive numbers that are not representable as a primitive sum of three nonzero squares.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 10, 12, 13, 15, 16, 20, 23, 24, 25, 28, 31, 32, 36, 37, 39, 40, 44, 47, 48, 52, 55, 56, 58, 60, 63, 64, 68, 71, 72, 76, 79, 80, 84, 85, 87, 88, 92, 95, 96, 100, 103, 104, 108, 111, 112, 116, 119, 120, 124, 127, 128, 130, 132, 135, 136, 140, 143, 144

Offset: 1

Wolfdieter Lang, Apr 06 2013

nonn

This is the complement of A223731. There an F. Halter-Koch reference is given.

			For a(1) up to a(7) there is no representation as sum of three nonzero squares.
a(8) = 12 because the only representation of 12 as a sum of nonzero squares is given by [a,b,c] = [2,2,2] = 2*[1,1,1], and this is not a primitive sum because gcd(2,2,2) = 2, not 1.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A223730, A223731, A223732, A223733, A223734.

notThreeSquaresQ[n_] := Select[ PowersRepresentations[n, 3, 2], Times @@ #1 != 0 && GCD @@ #1 == 1 & ] == {}; Select[Range[200], notThreeSquaresQ] (* Jean-François Alcover, Jun 21 2013 *)

a(n) has no representation as a^2 + b^2 + c^2 with 0 < a <= b <= c and gcd(a,b,c) = 1.

Conjectured g.f.: (2*x^61 -x^60 +2*x^59 -x^58 -2*x^57 +x^43 +3*x^42 -3*x^41 +x^40 -2*x^39 +2*x^32 -x^31 +2*x^30 -x^29 -2*x^28 +x^23 +3*x^22 -3*x^21 +x^20 -2*x^19 +x^18 +2*x^16 -3*x^14 +x^12 +3*x^11 -x^10 +x^6 -x^5 +x^4 +2*x^2 +x +1)*x / (x^4 -x^3 -x +1). - Alois P. Heinz, Apr 06 2013

cp's OEIS Frontend

A223731 All positive numbers that are primitive sums of three nonzero squares.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A223730 Multiplicities for representations of positive numbers n as primitive sums of three nonzero squares.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A223734 Positive numbers that are the sum of three nonzero squares with no common factor > 1 in exactly three ways.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A223735 Positive numbers that are not representable as a primitive sum of three nonzero squares.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula