A045864 Number of root quadruples with entry -n for integer Apollonian circle packings.

1, 1, 2, 2, 2, 3, 3, 3, 4, 3, 4, 6, 4, 5, 6, 5, 5, 7, 6, 6, 10, 7, 7, 10, 6, 7, 10, 10, 8, 10, 9, 9, 14, 9, 10, 14, 10, 11, 14, 10, 11, 18, 12, 14, 14, 13, 13, 18, 15, 11, 18, 14, 14, 19, 14, 18, 22, 15, 16, 20, 16, 17, 26, 17, 14, 26, 18, 18, 26, 18, 19, 26

Offset: 1

Author

Jeffrey C. Lagarias (lagarias(AT)umich.edu)

Keywords

• nonn
• nice
• look

Links

• Michel Marcus, Table of n, a(n) for n = 1..10000
• R. L. Graham, J. C. Lagarias, C. L. Mallows, Allan Wilks and C. H. Yan, Apollonian Circle Packings: Number Theory, arXiv:math/0009113 [math.NT], 2000-2003.
• R. L. Graham, J. C. Lagarias, C. L. Mallows, Allan Wilks and C. H. Yan, Apollonian Circle Packings: Number Theory, J. Number Theory, 100 (2003), 1-45.

Programs

Mathematica

chim4[p_] := If[p != 2, (-1)^((p - 1)/2), 0];
delta[n_] := If[Mod[n, 4] == 2, 1, 0];
a[n_] := If[n == 1, 1, n/4 Product[1 - chim4[p]/p, {p, FactorInteger[n][[All, 1]]}] + 2^(PrimeNu[n] - delta[n] - 1)];
Array[a, 72] (* Jean-François Alcover, Jan 26 2019, from PARI *)

PARI

chim4(p) = if (p % 2, (-1)^((p-1)/2), 0);
delta(n) = if ((n % 4)==2, 1, 0);
a(n) = {if (n==1, 1, f = factor(n)[,1]; n/4*prod(k=1, #f~, (1 - chim4(f[k])/f[k])) + 2^(omega(n)-delta(n)-1));} \\ Michel Marcus, May 13 2015

Formula

See Theorem 4.3 in Graham et al. link.

Extensions

Thanks to Robert G. Wilson v for pointing out that one of the terms was wrong.

Offset changed to 1 and more terms from Michel Marcus, May 13 2015