primes in pi: 1, 2, 6, 38, 16208, 47577, 78073, ...

[Russell Walsmith]

The idea of using the binary expansion of pi suggests another way to get a
sequence of primes:

Starting with the first digit of A004601 =
1,1,0,0,1,0,0,1,0,0,0,0,1,1,1,1,1,1,0,1,1,0,1,... we move rightward until we
encounter another 1. Since 11 (= 3 in decimal) is prime, we move to the next
1 and repeat the process.
11 = 3
1001001 = 73
1001000011111 = 4639
1000011 = 67
11 = 3
11 = 3
11 = 3

This gives the sequence 3, 73, 4639, 67, 3, 3, 3, 3, 3, 5, 3, 5, 5, 5, 17,
17, 1069...

Can anyone extend this?

Russell



On 7/23/06, N. J. A. Sloane <njas at research.att.com> wrote:
>
> Richard Guy said:
> > It would be more natural (??) to do this in base 2.
> >
> > So here are two new(?) sequences for people to
> > check and extend:
> >
> > The first
> >           2,  8,    14,    18,   ...
> >
> > digits in the binary representation of pi form the
> > primes
> >           3, 401, 25667, 410687, ...
> >
> > in decimal notation.
>
> Me:
>
> The binary expansion of Pi is A004601:
> 1,1,0,0,1,0,0,1,0,0,0,0,1,1,1,1,1,1,0,1,1,0,1,0,1,0,1,0,...
>
> Converting first n bits to an integer gives A068425:
> 1,3,6,12,25,50,100,201,402,804,1608,3216,...
>
> The primes here are A117721:
> 3,6588397,1686629713,26986075409,16703571626015105435307505830654230989,
> ...
>
> and they occur for these values of n (A065987):
> 2,23,31,35,124,323,2787,5717,6506 (and that's all I have)
> The latter sequence was computed by Bob Wilson.
>
> Eric, can you extend it?
>
> Richard, I seem to disagree with your results, but perhaps
> I misunderstood your message?
>
> Neil
>
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