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A223730 Multiplicities for representations of positive numbers n as primitive sums of three nonzero squares. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,33

COMMENTS

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.

REFERENCES

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

F. Halter-Koch, Darstellung natuerlicher Zahlen als Summe von Quadraten, Acta Arith.42 (1982) 11-20.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..10000

FORMULA

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.

EXAMPLE

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.

MAPLE

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^2<n, 0, b(n, i-1, t, select(x->x<i, s))+

      `if`(i^2>n, 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

MATHEMATICA

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

CROSSREFS

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

Sequence in context: A208249 A029422 A152800 * A117452 A029412 A178670

Adjacent sequences:  A223727 A223728 A223729 * A223731 A223732 A223733

KEYWORD

nonn

AUTHOR

Wolfdieter Lang, Apr 04 2013

STATUS

approved

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Last modified November 22 13:47 EST 2019. Contains 329393 sequences. (Running on oeis4.)