A278083 a(n) is 1/6 of the number of primitive integral quadruples with sum = 2*m and sum of squares = 2*m^2, where m = 2*n-1.

1, 4, 4, 8, 12, 12, 12, 16, 16, 20, 32, 24, 20, 36, 28, 32, 48, 32, 36, 48, 40, 44, 48, 48, 56, 64, 52, 48, 80, 60, 60, 96, 48, 68, 96, 72, 72, 80, 96, 80, 108, 84, 64, 112, 88, 96, 128, 80, 96, 144, 100, 104, 128, 108, 108, 144, 112, 96, 144, 128, 132, 160

Offset: 1

Author

Colin Mallows, Nov 14 2016

Keywords

• nonn

Comments

Set b(m) = a(n) for m = 2*n-1, and b(m) = 0 for m even.

Conjecture: b(m) is multiplicative: for k >= 1, b(2^k) = 0; b(p^k) = p^(k-1)*b(p) for p an odd prime; b(p) = p+1 for p == 3 (mod 4); b(p) = p-1 for p == 1 (mod 4). It would be nice to have a proof of this.

Examples

			6*a(2) = 24 = 6*b(3) because of (-1,2,2,3) and (0,1,1,4) (12 permutations each). For example, (-1) + 2 + 2 + 3 = 6 = 2*3 and (-1)^2 + 2^2 + 2^2 + 3^2 = 18 = 2*3^2 (with n = 2 and m = 3 = 2*n - 1).
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..500
• Petros Hadjicostas, Slight modification of Mallows' R program. [To get the total counts for n = 1 to 120, with the zeros, i.e., the sequence (b(n): n >= 1) shown in the comments above, type gc(1:120, 2, 2), where r = 2 and s = 2. To get the 1/6 of these counts with no zeros, type gc(seq(1,59,2), 2, 2)[,3]/6.]
• Colin Mallows, R programs for A278081-A278086.

Crossrefs

Cf. A046897, A278081, A278082, A278084, A278085, A278086.

Programs

Mathematica

sqrtint = Floor[Sqrt[#]]&;
q[r_, s_, g_] := Module[{d = 2 s - r^2, h}, If[d <= 0, d == 0 && Mod[r, 2] == 0 && GCD[g, r/2] == 1, h = Sqrt[d]; If[IntegerQ[h] && Mod[r + h, 2]==0 && GCD[g, GCD[(r+h)/2, (r-h)/2]]==1, 2, 0]]] /. {True -> 1, False -> 0};
a[n_] := Module[{m = 2n - 1, s}, s = 2m^2; Sum[q[2m - i - j, s - i^2 - j^2, GCD[i, j]] , {i, -sqrtint[s], sqrtint[s]}, {j, -sqrtint[s - i^2], sqrtint[s - i^2]}]/6];
Table[an = a[n]; Print[n, " ", an]; an, {n, 1, 100}] (* Jean-François Alcover, Sep 20 2020, after Andrew Howroyd *)

PARI

q(r, s, g)={my(d=2*s - r^2); if(d<=0, d==0 && r%2==0 && gcd(g, r/2)==1, my(h); if(issquare(d, &h) && (r+h)%2==0 && gcd(g, gcd((r+h)/2, (r-h)/2))==1, 2, 0))}
a(n)={my(m=2*n-1, s=2*m^2); sum(i=-sqrtint(s), sqrtint(s), sum(j=-sqrtint(s-i^2), sqrtint(s-i^2), q(2*m-i-j, s-i^2-j^2, gcd(i,j)) ))/6} \\ Andrew Howroyd, Aug 02 2018

Extensions

Terms a(51) and beyond from Andrew Howroyd, Aug 02 2018

Name and example section edited by Petros Hadjicostas, Apr 21 2020