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## Re: Brun's constants

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• (Shoot, I keep sending to the individual, not the group. This is the only list I am on that defaults to individual responses, rather than group responses.) A
Message 1 of 5 , May 28 8:29 AM
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(Shoot, I keep sending to the individual, not the group. This is the
only list I am on that defaults to individual responses, rather than
group responses.)

A different take on data analysis.

If you plot sum of inverses versus product((p-1)/(p-2)) you get an
almost perfect linear fit, with correlation coefficient r=1-(3*10^-
5). For n=50 data points that's a pretty good fit.

--- In primenumbers@yahoogroups.com, "Mark Underwood" <marku606@y...>
wrote:
>
> I just surveyed the gold mine of data which Adam presented below,
the
> quality of which went over and above what I was hoping for. And
even
> more that that I am elated that it so strongly corresponds with
what
> I was heuristically hanging my hat on. I was expecting somewhat
> tighter results but I am still very happy with these results which
> confirm my heuristics to a great degree of confidence. I have
grouped
> Adam's data to show how it falls into the pattern.
>
> To five significant digits (rounded),
>
> Primes separated by 2^n, that is 2, 4, 8, 16, 32, and 64 have
inverse
> sums of .017153, .017049, .017045, .016999, .016973 and .016981
> respectively. That is about .0170. Call this B' for Brun
> modified.
>
> Primes separated by (2^n) x (3^m), that is 6, 12, 18, 24, 36, 48,
54,
> 72 and 96 have inverse sums
>
of .033958, .034063, .034055, .034087, .034211, .034002, .034026, .03
> 3998, and .034012 respectively. That is about .0340. The
expected
> result is B' x (3-1)/(3-2) = .0340
>
> Primes separated by (2^n) x (5^m), that is 10, 20, 40, 50, 80, and
> 100 have inverse sums
> of .022674, .022699, .022724, .022635, .022733, and .022683
> respectively. That is about .0227. The expected result is B' x (5-
1)/
> (5-2) = .0227
>
> Primes separated by (2^n) x (7^m), that is 14, 28, 56, and 98 have
> inverse sums of .020386, .020450, .020338 and 020352 respectively.
> That is about .0204. The expected result is B' x ((7-1)/(7-2)
> = .0204.
>
> Primes separated by (2^n) x (11^m), that is 22, 44 and 88 have
> inverse sums of .018834, .018879 and .018920 respectively. That is
> about .0189. The expected result is B' x (11-1)/(11-2) = .0189.
>
> Primes separated by (2^n) x (13^m), that is 26 and 52 have inverse
> sums of .018643 and .018612 respectively. That is about .0186. The
> expected result is B' x (13-1)/(13-2) = .0185.
>
> And so on.
>
> Primes separated by (2^n) x (3^m) x (5^r), that is 30, 60 and 90
have
> inverse sums of .045331, .045362 and .045378 respectively. That is
> about .0453. The expected result is B' x (3-1)/(3-2) x (5-1)/(5-2)
> = .0453.
>
> Primes separated by (2^n) x (3^m) x (7^r), that is 42 and 84 have
> inverse sums of .040838 and .040786. That is about .0408. The
> expected result is B' x (3-1)/(3-2) x (7-1)/(7-2) = .0408
>
> Primes separated by (2^n) x (5^m) x (7^r), that is 70 has an
inverse
> sum of about 0273. The expected result is B' x (5-1)/(5-2) x (7-1)/
(7-
> 2) = .272
>
> Good enough for me! Thanks Adam!
>
> Mark
>
>
>
>
> --- In primenumbers@yahoogroups.com, "Adam" <a_math_guy@y...> wrote:
> > Hmmm, I posted some data (somewhere) but it isn't showing up on
the
> > list. Did I send it to somebody's email, instead of the list?
If
> > so, please post for me. Anyway, it defintely looked like there
> were
> > patterns (wait, I might have saved the data)......here ya' go,
sum
> of
> > (1/p) for p>10^6 and p<20 million when (p+k) is also prime,
output
> is
> > (k,sum):
> >
> >
> > 2, .01715295754
> > 4, .01704852064
> > 6, .03395755937
> > 8, .01704518900
> > 10, .02267391029
> > 12, .03406331599
> > 14, .02038594306
> > 16, .01699946972
> > 18, .03405521233
> > 20, .02269903011
> > 22, .01883421188
> > 24, .03408741833
> > 26, .01864293196
> > 28, .02044996718
> > 30, .04533148052
> > 32, .01697346702
> > 34, .01813592253
> > 36, .03421070761
> > 38, .01801860557
> > 40, .02272393752
> > 42, .04083825580
> > 44, .01887947490
> > 46, .01780359233
> > 48, .03400242989
> > 50, .02263451790
> > 52, .01861219959
> > 54, .03402561886
> > 56, .02033752356
> > 58, .01760087352
> > 60, .04536210827
> > 62, .01762533361
> > 64, .01698090345
> > 66, .03778604223
> > 68, .01808951646
> > 70, .02731960955
> > 72, .03398823253
> > 74, .01746662755
> > 76, .01806215382
> > 78, .03717973980
> > 80, .02273260129
> > 82, .01748673074
> > 84, .04078555194
> > 86, .01747334842
> > 88, .01892019678
> > 90, .04537844859
> > 92, .01780730452
> > 94, .01745072896
> > 96, .03401206908
> > 98, .02035243350
> > 100, .02268286529
> >