Re Primes with initial digits - errors: one seq. left to check

[Martin Fuller]

Actually I think that A068120 has the occasional
downward step.  But the first one isn't expected until
around 10^8 so I'm not looking for it.

We can estimate the 'probability' that A068120(n) >
A068120(n+1) using the lengths of the prefixes and the
density of primes.  Then we can get the 'expected'
number of downward steps by summing the probabilities.

Suppose n and n+1 have the same number of base-10
digits, d, and the required prefixes are x and y.  The
requirement for A068120(n) > A068120(n+1) is:
10 composites above x*10, and
10^2 composites above x*10^2, and
...
10^(d+1) composites above x*10^(d+1), and
  either:
  a prime amongst 10 numbers above y*10
  10^(d+2) composites above x*10^(d+2), and
    either:
    a prime amongst 10^2 numbers above y*10^2
    ...

Some shorthand:
P(n,d) = probability that A(n) > A(n+1), with n and
n+1 having d digits.

Q(a,b) = probability that 10^b numbers around 10^a are
composite.

R(a,b,d) = probability of 10^(b+d) composites around
10^a, and either:
  a prime amongst an independent 10^b around 10^a, or
  R(a+1,b+1,d)


I haven't found a closed solution, but here it is
numerically in PARI/GP:

Q(a,b)=(1 - 1/log(10)/a)^(10^b)

/* We need to avoid infinite loops in the following:
R(a,b,d)=Q(a,b+d) * (1 - Q(a,b) * (1 - R(a+1,b+1,d)))
*/
R(a,b,d,z=10^-100)=local(q); q=Q(a,b+d); if(q < z, q,
q * (1 - Q(a,b) * (1 - R(a+1,b+1,d))))

P(n,d)=prod(i=1, d, Q(n*d+log(n/10^d)+i,i)) *
R(n*d+log(n/10^d)+d+1, 1, d)

sum(d=3,7, intnum(n=10^(d-1),10^d-1, P(n,d)))
/* 0.96 */
sum(d=3,8, intnum(n=10^(d-1),10^d-1, P(n,d)))
/* 1.20 */

--- Dan Dima <dimad72 at gmail.com> wrote:

> I can confirm these results. Some more terms:
> 
> 21212121212121212121212121212121212121212119
> 2222222222222222222222222222222222222222222297
> 232323232323232323232323232323232323232323232361
> 242424242424242424242424242424242424242424242424293
>
25252525252525252525252525252525252525252525252525061
>
2626262626262626262626262626262626262626262626262626003
>
2727272727272727272727272727272727272727272727272727271
>
28282828282828282828282828282828282828282828282828282828057
>
292929292929292929292929292929292929292929292929292929292907
>
303030303030303030303030303030303030303030303030303030303030043
>
3131313131313131313131313131313131313131313131313131313131313161
>
3232323232323232323232323232323232323232323232323232323232323232007
>
333333333333333333333333333333333333333333333333333333333333333333053
>
3434343434343434343434343434343434343434343434343434343434343434343431
>
3535353535353535353535353535353535353535353535353535353535353535353535127
> 
> 
> I think we can conjecture that A068120 is a strictly
> increasing sequence...
> 
> An interesting subsequence of this might be that one
> that has exactly one
> extra digit appended...
> 
> Meanwhile I think that A088639 is redundant since
> A088639(n)=A068120(n).
> 
> A068008 is somehow tricky if we consider leading
> 0's:
> A068008(25)=153
> but:
>
25252525252525252525252525252525252525252525252525061
> is a smaller prime than:
>
25252525252525252525252525252525252525252525252525163
> A068008(26)=163
>  but:
>
2626262626262626262626262626262626262626262626262626003
> is a smaller prime than:
>
2626262626262626262626262626262626262626262626262626163
> 
> 
> Best regards,
> Dan Dima
> 
> 
> 
> 
> 
> 
> 
> 
> 
> On 1/27/07, Jacques Tramu
> <jacques.tramu at echolalie.com> wrote:
> >
> >
> >
> > >From: "N. J. A. Sloane" <njas at research.att.com>
> > > could someone please check it?  Here is the
> current entry:
> >
> > >
> > > %I A068120
> > > %S A068120
> > >
>
13,223,3331,44449,555557,66666629,777777701,888888883,99999999907,
> > > %T A068120
> 1010101010101010101039,11111111111111111111111,
> > > %U A068120
>
12121212121212121212121223,1313131313131313131313131327
> > > %N A068120 Smallest prime beginning with exactly
> n n's.
> > > %K A068120 base,easy,nonn,new
> > > %O A068120 1,1
> > > %A A068120 Amarnath Murthy
> (amarnath_murthy(AT)yahoo.com), Feb 21 2002
> > > %E A068120 Edited and extended by Robert G.
> Wilson v (rgwv(AT)rgwv.com),
> > > Feb 22 2002
> > > %E A068120 a(7) corrected by Amarnath Murthy
> > > (amarnath_murthy(AT)yahoo.com), Mar 24 2002
> > > %E A068120 Corrected by T. D. Noe
> (noe(AT)sspectra.com), Nov 14 2006
> > >
> >
> > Here are my results  + some new entries - 
> different entry  is flagged (*)
> >
> > 13 , 223, 3331, 44449, 555557, 66666629,
> 777777701, 888888883,
> > 99999999907,
> > 1010101010101010101039, 11111111111111111111111,
> > 12121212121212121212121223,
> *1313131313131313131313131301*,
> > 141414141414141414141414141497,
> 15151515151515151515151515151501,
> > 1616161616161616161616161616161691,
> 171717171717171717171717171717171737,
> >
> >
>
181818181818181818181818181818181818059,1919191919191919191919191919191919191909,
> > 202020202020202020202020202020202020202093
> >
> > regards,
> > JT
> >
> >
>