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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
1, 6, 4, 9, 3, 3, 7, 6, 8, 9, 0, 9, 8, 0, 3, 0, 7, 0, 1, 0, 2, 5, 9, 4, 2, 9, 3, 3, 3, 6, 0, 1, 7, 8, 9, 6, 3, 6, 6, 9, 2, 3, 5, 7, 6, 6, 2, 5, 6, 6, 1, 1, 4, 4, 9, 0, 5, 7, 7, 2, 4, 8, 8, 3, 8, 4, 2, 5, 6, 4, 5, 1, 8, 9, 4, 8, 0, 7, 7, 2, 5, 2, 0, 6, 9, 0, 2, 0, 4, 2, 4, 8, 5, 2, 5, 3, 6, 0, 1, 0, 2, 7, 0, 1, 7 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Table of n, a(n) for n=1..105.

Steven R. Finch, Apollonian circles with integer curvatures, p. 6. [Cached copy, with permission of the author]

Elena Fuchs and Katherine Sanden, Some experiments with integral Apollonian circle packings, arXiv:1001.1406 [math.NT] p. 7.

EXAMPLE

1.64933768909803...

From Jon E. Schoenfield, Nov 23 2018: (Start)

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

        k       p(k)  P(k)

  =======  =========  ==============================

        1          3  1.6666666666666666666666666...

        2          7  1.6534391534391534391534391...

        4         19  1.6498966974113172943582300...

        8         47  1.6494053597054879136371001...

       16        127  1.6493477221030105342383210...

       32        283  1.6493391548877836162710884...

       64        683  1.6493379309967496039826803...

      128       1567  1.6493377307932024281500980...

      256       3607  1.6493376964319913937637047...

      512       8111  1.6493376904381138856289227...

     1024      17579  1.6493376893541631519325267...

     2048      38699  1.6493376891481337717879033...

     4096      83639  1.6493376891079951056875450...

     8192     180331  1.6493376891000508253007793...

    16384     385531  1.6493376890984468938333617...

    32768     820163  1.6493376890981175839368748...

    65536    1741379  1.6493376890980490243352558...

   131072    3679183  1.6493376890980346056857763...

   262144    7750943  1.6493376890980315407340724...

   524288   16284787  1.6493376890980308829733718...

  1048576   34128323  1.6493376890980307407240351...

  2097152   71367371  1.6493376890980307097441549...

  4194304  148939543  1.6493376890980307029516022...

  8388608  310236419  1.6493376890980307014534855...

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

MATHEMATICA

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

CROSSREFS

Cf. A002145, A052483, A189226, A189227.

Sequence in context: A284149 A096499 A179258 * A021158 A231535 A019931

Adjacent sequences:  A248927 A248928 A248929 * A248931 A248932 A248933

KEYWORD

nonn,cons,changed

AUTHOR

Jean-Fran├žois Alcover, Oct 17 2014

EXTENSIONS

More digits from Vaclav Kotesovec, Jun 27 2020

STATUS

approved

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Last modified July 11 01:46 EDT 2020. Contains 335600 sequences. (Running on oeis4.)