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

%I A045864 #24 Jan 27 2019 04:36:45
%S A045864 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,
%T A045864 14,9,10,14,10,11,14,10,11,18,12,14,14,13,13,18,15,11,18,14,14,19,14,
%U A045864 18,22,15,16,20,16,17,26,17,14,26,18,18,26,18,19,26
%N A045864 Number of root quadruples with entry -n for integer Apollonian circle packings.
%H A045864 Michel Marcus, <a href="/A045864/b045864.txt">Table of n, a(n) for n = 1..10000</a>
%H A045864 R. L. Graham, J. C. Lagarias, C. L. Mallows, Allan Wilks and C. H. Yan, <a href="http://arxiv.org/abs/math/0009113">Apollonian Circle Packings: Number Theory</a>, arXiv:math/0009113 [math.NT], 2000-2003.
%H A045864 R. L. Graham, J. C. Lagarias, C. L. Mallows, Allan Wilks and C. H. Yan, <a href="https://doi.org/10.1016/S0022-314X(03)00015-5">Apollonian Circle Packings: Number Theory</a>, J. Number Theory, 100 (2003), 1-45.
%F A045864 See Theorem 4.3 in Graham et al. link.
%t A045864 chim4[p_] := If[p != 2, (-1)^((p - 1)/2), 0];
%t A045864 delta[n_] := If[Mod[n, 4] == 2, 1, 0];
%t A045864 a[n_] := If[n == 1, 1, n/4 Product[1 - chim4[p]/p, {p, FactorInteger[n][[All, 1]]}] + 2^(PrimeNu[n] - delta[n] - 1)];
%t A045864 Array[a, 72] (* _Jean-François Alcover_, Jan 26 2019, from PARI *)
%o A045864 (PARI) chim4(p) = if (p % 2, (-1)^((p-1)/2), 0);
%o A045864 delta(n) = if ((n % 4)==2, 1, 0);
%o A045864 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
%K A045864 nonn,nice,look
%O A045864 1,3
%A A045864 Jeffrey C. Lagarias (lagarias(AT)umich.edu)
%E A045864 Thanks to _Robert G. Wilson v_ for pointing out that one of the terms was wrong.
%E A045864 Offset changed to 1 and more terms from _Michel Marcus_, May 13 2015