

A045864


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


4



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
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OFFSET

1,3


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], 20002003.
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), 145.


FORMULA

See Theorem 4.3 in Graham et al. link.


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] (* JeanFrançois Alcover, Jan 26 2019, from PARI *)


PROG

(PARI) chim4(p) = if (p % 2, (1)^((p1)/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


CROSSREFS

Sequence in context: A064515 A112754 A283467 * A072302 A165360 A340542
Adjacent sequences: A045861 A045862 A045863 * A045865 A045866 A045867


KEYWORD

nonn,nice,look


AUTHOR

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


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


STATUS

approved



