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A248930 Decimal expansion of c = 2*Product_{prime p == 3 (mod 4)} (1 - 2/(p*(p-1)^2)), a constant related to the problem of integral Apollonian circle packings. 2

%I

%S 1,6,4,9,3,3,7,6,8,9,0,9,8,0,3

%N Decimal expansion of c = 2*Product_{prime p == 3 (mod 4)} (1 - 2/(p*(p-1)^2)), a constant related to the problem of integral Apollonian circle packings.

%H Steven R. Finch, <a href="/A189227/a189227.pdf">Apollonian circles with integer curvatures</a>, p. 6. [Cached copy, with permission of the author]

%H Elena Fuchs and Katherine Sanden, <a href="http://arxiv.org/abs/1001.1406">Some experiments with integral Apollonian circle packings</a>, arXiv:1001.1406 [math.NT] p. 7.

%e 1.64933768909803...

%e From _Jon E. Schoenfield_, Nov 23 2018: (Start)

%e Let p(j) be the j-th prime that is congruent to 3 (mod 4), i.e., p(j) = A002145(j), and let P(k) be the partial product 2*Product_{j=1..k} (1 - 2/(p(j)*(p(j)-1)^2)); then at k = 1, 2, 4, 8, ..., 2^23, we have

%e k p(k) P(k)

%e ======= ========= ==============================

%e 1 3 1.6666666666666666666666666...

%e 2 7 1.6534391534391534391534391...

%e 4 19 1.6498966974113172943582300...

%e 8 47 1.6494053597054879136371001...

%e 16 127 1.6493477221030105342383210...

%e 32 283 1.6493391548877836162710884...

%e 64 683 1.6493379309967496039826803...

%e 128 1567 1.6493377307932024281500980...

%e 256 3607 1.6493376964319913937637047...

%e 512 8111 1.6493376904381138856289227...

%e 1024 17579 1.6493376893541631519325267...

%e 2048 38699 1.6493376891481337717879033...

%e 4096 83639 1.6493376891079951056875450...

%e 8192 180331 1.6493376891000508253007793...

%e 16384 385531 1.6493376890984468938333617...

%e 32768 820163 1.6493376890981175839368748...

%e 65536 1741379 1.6493376890980490243352558...

%e 131072 3679183 1.6493376890980346056857763...

%e 262144 7750943 1.6493376890980315407340724...

%e 524288 16284787 1.6493376890980308829733718...

%e 1048576 34128323 1.6493376890980307407240351...

%e 2097152 71367371 1.6493376890980307097441549...

%e 4194304 148939543 1.6493376890980307029516022...

%e 8388608 310236419 1.6493376890980307014534855...

%e with P(k) approaching 1.6493376890980307010... (End)

%t kmax = 25; Do[ P[k] = Product[p = Prime[n]; If[Mod[p, 4] == 3, 1 - 2/(p*(p - 1)^2) // N[#, 40]&, 1], {n, 1, 2^k}]; Print["P(", k, ") = ", P[k]], {k, 10, kmax}]; c = 2*P[kmax]; RealDigits[c, 10, 15] // First

%Y Cf. A002145, A052483, A189226, A189227.

%K nonn,cons,more

%O 1,2

%A _Jean-Fran├žois Alcover_, Oct 17 2014

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Last modified October 14 18:28 EDT 2019. Contains 328022 sequences. (Running on oeis4.)