Some thoughts on A089338

[Chuck Seggelin]

Hello Seqfans,

I was testing out an idea this weekend and ended up reproducing a sequence
that is already in the OEIS:

============================================
ID Number: A089338
URL:       http://www.research.att.com/projects/OEIS?Anum=A089338
Sequence:  3,1,1,2,1,6,11,16,11,1,8,21,13,11,34,41,12,4,66,24,15,17,4,
           122,70,96,33,2,43,5,3,100,44,28,23,27,12,4,113,10,3,90,9,
           162,15,9,69,146,9,145,74,3,42,99,31,93,35,259,53,79,14,285,
           84,1,36,78,147,78,66,246,155,624
Name:      Beginning with 3 the smallest number such that the concatenation
a(n),
              a(n-1), ... a(2), a(1) is a prime.
Example:   3, 13, 113, 2113, 12113, 612113, ...etc are prime.
See also:  Adjacent sequences: A089335 A089336 A089337 this_sequence A089339
              A089340 A089341
           Sequence in context: A010275 A080847 A095276 this_sequence
A073166
              A050169 A064048
Keywords:  base,nonn
Offset:    0
Author(s): Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Nov 04 2003
Extension: More terms from Ray Chandler (RayChandler(AT)alumni.tcu.edu), Nov
              07 2003
============================================

There are inifinitely many such sequences, each starting at a different
prime.  For example, for the starting prime 7 the sequence would be:

7, 1, 3, 6, 2, 1, 2, 7, 3, 5, 12, 15, 4, 23, 22, 15, 21, 42, 36, 51, 14, 10,
39, 5, 61, 80, 66, 19, 14, 60, 108, 60, 46, 21, 62, 37, 201, 59, 15, 160,
51, 17, 67, 54, 6, 75, 8, 151, 113, ...

For 11 it would be:

11, 2, 4, 3, 2, 4, 15, 3, 9, 20, 4, 3, 11, 31, 6, 24, 23, 82, 11, 21, 3, 22,
20, 63, 19, 56, 22, 17, 42, 105, 31, 2, 4, 27, 96, 42, 5, 72, 19, 20, 22,
32, 102, 31, 104, 4, 24, 95, 21, 13, 12, 9, 38, 3, 58, 38, 78, 31, 119, 31,
45, 107, 42, 12, 9, 21, 66, 181, 357, 200, 7, 152, 82, 9, 116, 84, 211, 660,
131, 75, 135, 508, 2, 40, 324, 38, 136, 39, 12, 212, 279

There is probably no reason to include these two sequences in the OEIS
however, because only the first sequence where a(0)=3 is necessary to
capture the idea.  But I wonder: is there any interesting way to capture
information about all such sequences?

I note that in A089338, the addition of only a single digit was required to
get terms A089338(1) to A089338(5).  To get term A089338(6) two digits are
required.  In the sequence starting at 7, prepending single digits yields
terms 1 to 9.

This gave me an idea for 3 sequences:

a000001(n) = beginning with the n'th prime, the number of times a new prime
is formed by prepending the smallest nonzero digit.  For example a(2) is 5
because the second prime is 3, to which single digits can be prepended 5
times yielding a new prime each time--13, 113, 2113, 12113, 612113.  (There
is no nonzero digit which can be prepended to 612113 to yield a new prime.)

0, 5, 0, 9, 5, 4, 8, 4, 5, 9, 4, 6, 2, 7, 6, 8, 9, 7, 6, 3, 14, 5, 5, 2, 4,
10, 1, 5, 7, 3, 4, 3, 5, 5, 0, 6, 5, 8, 5, 13, 4, 5, 4, 5, 3, 8, 4, 4, 5, 8,
3, 6, 1, 4, 4, 2, 5, 2, 2, 3, 4, 9, 8, 7, 4, 7, 3, 3, 5, 5, 7, 8, 4, 3, 3,
2, 1, 7, 0, 4, 3, 5, 3, 7, 9, 6, 6, 5, 6, 8, ...

a000002(n) = the values in a000001 which are records (all primes up to
287117 tested):

0, 5, 9, 14, 15, ...

a000003(n) = the first prime yielding the record value a000002(n)

2, 3, 7, 73, 13799, ...

Possibly one could include a fourth sequence showing the primes produced by
prepending single digits for each of the record holders: 2, 612113,
5372126317, 4818372912366173, 21291981879276213799.

The description of these sequences would need to be cleaned up, but I'm
curious if anyone thinks they are worth adding to the OEIS?

One final note of interest about the prime produced by 73
(4818372912366173)... 73 itself is produced by adding a single digit to the
prime 3. Thus 4818372912366173 is a left-truncatable prime.

If you don't allow primes containing the digit zero, the list of
left-truncatable primes is finite, right?  There are a little over 4,000 of
them, and the largest is 24 digits long.  Though the analysis which produced
the sequences above doesn't allow the addition of 0 to the starting prime,
any starting prime is allowed, even those which contain zeros.  So in
theory, there may well be primes to which more than 23 single nonzero digits
can be prepended yielding a prime each time.

-- Chuck Seggelin