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%I #20 Mar 20 2024 09:40:25
%S 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,
%T 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,
%U 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
%N Multiplicities for representations of positive numbers n as primitive sums of three nonzero squares.
%C 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).
%C 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.
%C 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.
%C 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.
%C 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.
%D E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985.
%H Alois P. Heinz, <a href="/A223730/b223730.txt">Table of n, a(n) for n = 1..10000</a>
%H F. Halter-Koch, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa42/aa4212.pdf">Darstellung natürlicher Zahlen als Summe von Quadraten</a>, Acta Arith. 42 (1982) 11-20.
%F 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.
%e 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.
%e 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.
%p with(numtheory):
%p b:= proc(n, i, t, s) option remember;
%p `if`(n=0, `if`(t=0 and s={}, 1, 0), `if`(i=1, `if`(t=n, 1, 0),
%p `if`(t*i^2<n, 0, b(n, i-1, t, select(x->x<i, s))+
%p `if`(i^2>n, 0, b(n-i^2, i, t-1, `if`(s={1}, factorset(i),
%p s intersect factorset(i)))))))
%p end:
%p a:= n-> b(n, isqrt(n), 3, {1}):
%p seq(a(n), n=1..200); # _Alois P. Heinz_, Apr 06 2013
%t a[n_] := Select[ PowersRepresentations[n, 3, 2], Times @@ # != 0 && GCD @@ # == 1 &] // Length; Table[a[n], {n, 1, 134}] (* _Jean-François Alcover_, Jun 21 2013 *)
%Y Cf. A223731, A025427 (non-primitive case), A223732, A223733, A223734.
%K nonn
%O 1,33
%A _Wolfdieter Lang_, Apr 04 2013
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