Identical-digit blocks in decimal representation of partition numbers

[Jonathan Post]

Why (statistically and the well-known facts of powers of 5 and 6 when
represented base 10)  is that when I scan Plouffe's table for the string
"78125" (i.e. 5^7) I find 17 hits, of which 2 have the string at the
right-hand end: P(2694) and P(5224).  In both cases, we have divisibility by
5^5 (but not by higher power of 5).

When I scan for the string "46656" (i.e. 6^6) there are 15 hits, none at
right end. Scanning for "7776" (i.e. 6^5) there are 149 hits, of which only
1 is at the right-hand namely P(9132) and two at left-end (3760 and 13380 of
which the latter also has the string in the middle as a double-hit). P(9132)
is divisible by 2^6 * 3 but no higher power of 2 or 3.


On 8/10/06, Jonathan Post <jvospost3 at gmail.com> wrote:
>
>  Ralf makes sense here.  Consecutive digits matter in Ramanujan-type
> congruences specifically where blocks of K zeroes occur at the right-hand
> end of the digit string, i.e. when P(n) is a multiple of 10^K. Using the
> extended list of Plouffe (through n = 16456) we find:
>
> 101 values of P(n) with blocks of eactly 4 zeroes, of which 11 occur with
> the zeroes at the right (599, 776, 1949, 2499, 4989, 7964, 8249, 12499,
> 12624, 14574, 16274); there appearing to be some structure here (i.e. 2499
> + 10000 = 12499).
>
> 10 values of P(n) with blocks of eactly 5 zeroes, of which 1 occurs with
> the zeroes at the right (11224).
>
> There are well-known P(n) mod 5 congruences; the powers of 10 are relevant
> to simultaneously mod 2 and mod 5.
>
>
>
>
> On 8/9/06, Ralf Stephan <ralf at ark.in-berlin.de> wrote:
> >
> > > Me:  Am I the only seqfan who finds this kind of investigation
> > > extremely repugnant?
> >
> > You're not. In this case, however, it could be a hint for more
> > Ramanujan-type congruences. The problem is that this is not followed
> > up by more serious research into the matter.
> >
> >
> > Regards,
> > ralf
> >
> >
>
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