A224444 Multiplicities for representations of nonnegative numbers as primitive sums of three squares of integers (square 0 allowed).

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

Offset

0,18

Comments

a(n) = 0, for n >= 1, if there is no representation of n as a sum of three squares (square 0 allowed) with no common factor > 1. a(0) = 0 because gcd(0,0,0) = 0 (not 1). a(n) = k >= 1 if n is representable as a primitive sum of three squares (square 0 allowed) in exactly k ways, if neither the order of the three terms nor the signs of the numbers to be squared are taken into account.

Compare with the multiplicities A000164.

The numbers for which a(n) is not 0 are given in A047449.

Links

Table of n, a(n) for n=0..125.

Formula

a(n) = k if n, for n >= 0, has exactly k representations n = a^2 + b^2 + c^2, with a, b and c integers, 0 <= a <= b < = c and gcd(a,b,c) = 1. If there is no such representation a(n) = 0.

Example

a(0) = 0 because  0 = 0^2 + 0^2 + 0^2 is the only candidate for a representation but this is not a primitive sum because gcd(0,0,0) = 0, not 1.
a(2) = 1 because the only candidate for a representation of 2 is the triple [a,b,c] = [0,1,1] and this is primitive, because gcd(0,1,1) = 1.
a(9) = 1 because the two candidate triples are [0, 0, 3] and [1, 2, 2] but [0, 0, 3] is not primitive (gcd(0,0,3) =  3). A000164(9) = 2.
a(17) = 2 with the primitive [a,b,c] triples [0, 1, 4] and [2, 2, 3]. A000164(17) = 2 also.
a(41) = 3 = A000164(41) because the candidate triples [0, 4, 5], [1, 2, 6] and [3, 4, 4] are all primitive.

Mathematica

Table[ Count[ PowersRepresentations[n, 3, 2], pr_ /; GCD @@ pr == 1], {n, 0, 125}] (* Jean-François Alcover, Apr 09 2013 *)

Crossrefs

Cf. A047449, A000164.

Sequence in context: A133699 A157361 A369454 * A101808 A145865 A341281

Adjacent sequences: A224441 A224442 A224443 * A224445 A224446 A224447

Keyword

nonn

Author

Wolfdieter Lang, Apr 08 2013

Status

approved