A189227 Primes among the curvatures in the nickel-dime-quarter Apollonian circle packing A189226.

-11, 61, 157, 181, 349, 373, 397, 421, 541, 661, 709, 733, 829, 853, 877, 997, 1021, 1069, 1093, 1213, 1237, 1381, 1429, 1597, 1621, 1669, 1693, 1741, 1861, 2029, 2221, 2293, 2341, 2389, 2557, 2677, 2749, 2917, 3037, 3061, 3109, 3181, 3229, 3253, 3301, 3373

Offset: 1

Author

Jonathan Sondow, Apr 22 2011

Keywords

• sign

Comments

See A189226 for comments, references, links, examples, and crossrefs.

Links

• Giovanni Resta, Table of n, a(n) for n = 1..1128 terms < 10^5.
• S. Finch, Apollonian circles with integer curvatures [Cached copy, with permission of the author]

Crossrefs

Cf. A248930, A248938.

Programs

Mathematica

(* terms < 10^4 *) t = Range[9999]*0; w = {-11, 21, 24, 28}; s[1] = {{-1,2,2,2}, {0,1,0,0}, {0,0,1,0}, {0,0,0,1}}; s[2] = {{1,0,0,0}, {2,-1,2,2}, {0,0,1,0}, {0,0,0,1}}; s[3] = {{1,0,0,0}, {0,1,0,0}, {2,2,-1,2}, {0,0,0,1}}; s[4] = {{1,0,0,0}, {0,1,0,0}, {0,0,1,0}, {2,2,2,-1}}; r[m_, j_, p_] := Block[{v = (m.w)[[p]]}, If[v < 9999, t[[v]] = 1; Do[ If[i != j, r[m.s[i], i, p]], {i, 4}]]]; Do[ r[s[i], i, i], {i, 4}]; Prepend[ Select[ Flatten@ Position[t,1], PrimeQ], -11] (* Giovanni Resta, Jan 02 2014 *)

Formula

a(n) == 13 (mod 24) (because a(n) is prime, a(n) = A189226(k) for some k, and all terms of A189226 are == 0, 4, 12, 13, 16, or 21 (mod 24)).

Extensions

Corrected and extended by Steven Finch, Jan 02 2014

a(16)-a(46) from Giovanni Resta, Jan 02 2014