Possible Comment on A007095

Tue May 22 21:42:48 CEST 2007

Neil,

    I am waiting for my new computer which should have the needed
horse power to extend these sequences. See {A082830, A082831, A082832,
A082833, A082834, A082835, A082836, A082837, A082838, A082839 
</%7Enjas/sequences/A082839>}:
Decimal expansion of the (finite) value of the sum_{ k >= 1, k has no
{1..0} digit in base 10 } 1/k.

</%7Enjas/sequences/A082839>Sequentially yours, Bob.

N. J. A. Sloane wrote:

>Jeremy, i think the [sequence of decimal digits of the]
>number is in the OEIS.
>Neil
>
>Let S be the set of positive integers that, when written in
>base 10, does not contain the digit 9. Show that
>the sum of 1/n over all n = =91 S converges and is less than 80.
>
>
>As I recall, Bob Wilson (rgwv) was involved in
>computing it, or extending it. - Bob?
>
>Neil
>
>  
>

Dear Seqfans,

As I follow the above correspondance, I have just received an invitation
from Amazon to purchase "Evolutionary Computer Music "
by Eduardo Miranda. The description reads "Musicians, perhaps more than
any other class of artists, have always been acutely aware of the
thousand years ago of the direct relationship between the pitch of a
note and the length of a string or pipe, to the latest computer models
of human musical cognition and intelligence, musicians have always
looked to science to provide new and challenging ways to study and
compose music. With the great scientific advances being made in the
field of evolutionary biology, from new insights into the origins of
humans and other species to complete mappings of the genes that control
our growth and development, a new approach to the study of music is
chapters on the interplay between Evolutionary Computation and Music by
leading pioneers in the field."

May be worth a look.
Paul

Paul Barry
Head of School of Science
Waterford Institute of Technology, WIT
Cork Road, Waterford, Ireland
Tel +353-51-302027, fax +353-51-302679
email pbarry at wit.ie

Hmm... One place to start is with the easy stuff; for instance, due to the
figurate nature of squares it's easy to show that 1,4,9,16, etc. are in
A129783. All squares can be written:

1+3 = 4
1+3+5 = 9
1+3+5+7 = 16
1+3+5+7+9 = 25
etc.

Since the sum of any set of constant-spaced integers greater than two is a
composite integer, and those sets are easily defined via functions, we can

n^2 - 1 = pq = {3+5+7, 3+5+7+9, 3+5+7+9+11 ...} = n(n+2), so the product of
twin primes which differ by two are a subset this set;

n^2 - 4 = pq = {5+7+9, 5+7+9+11, 5+7+9+11+13 ...} = n(n+4), so the product
of twin primes which differ by four are a subset this set;

n^2 - 9 = pq = (7+9+11, 7+9+11+13, 7+9+11+13+15 ...) = n(n+6), so the
product of twin primes which differ by six are a subset this set;

and so on. for n^2 - r = pq the first value for r = 3 is:

9 - 3 = 6 = 2*3

Is this the only value where r = 3? I tend to think that the nonsquare
values for r are there because they're true in only a handful of cases. And,
there may be some value for n^2 above which there are no more cases for

-----Original Message-----

--------------------------------------------------
Warning:
Ignorant newbie e-mail follows;
reader discretion is advised.    ;-)
--------------------------------------------------
All,

I don't yet have a proof about any of the conjecturally-missing numbers in
A129783, but I did get (after using the first 283 primes, i.e., the primes
up through 1847) a list identical to the one currently given in the OEIS.
The eight numbers that required the largest primes were all perfect squares:

n =  16 was first found at p =   89, q =   97
n =  25 was first found at p =  139, q =  149
n =  36 was first found at p =  199, q =  211
n =  81 was first found at p =  523, q =  541
n = 100 was first found at p =  887, q =  907
n = 121 was first found at p = 1129, q = 1151
n = 144 was first found at p = 1669, q = 1693
n =  64 was first found at p = 1831, q = 1847

In each case, n = ((q - p) / 2)^2 , so p * q + n = ((p + q) / 2)^2.

That much kinda made sense to me ....  But I'm curious about some of the
other patterns I've noticed.  In particular, in looking at the numbers that
do and (conjecturally) don't occur in A129783, I'm wondering if there's some

are congruent to 26 mod 36.  Among all the integers in the range from 278
through 926, barely a third are in A129783, but every integer congruent to
26 mod 36 in that range is in A129783.

In the 1000 characters on the lines below (36 per line, except the last),
the X's represent numbers in A129783 (starting with 1), and the periods
represent numbers that (conjecturally) aren't in the sequence.  If the lines

are displayed in a fixed-pitch font (e.g., Courier New), the density of X's
in column 26 seems striking (to me, anyway <g>)!

X.XX....XX...X.X..X.X.X.XX..XX...XXX
.X....XX.XX.X...X....X...X.XXXXX....
..X.XX..XXX.XX.....X.XX....X..X..X..
XX..X.X.XXX.XX..X...X....X...XXX..XX
.X.XX....X......XXXX..X.X...XX....X.
XX.....X.X.....X......X..X...X.XX.X.
.XXXXX..X.X.....X...X.X.X...X.X..X..
XX.XX....XX.....X....X...X....XX....
XX.XXX..XX...X.....XX....X..XX....XX
.X...X.X..X.....X....XX..X..XXX.X...
XX.XX....X...X.....X....XX.....X.XX.
XX.X.X.X.XX.....X.....X..X..XXXX....
X.X.X...XX...XX.X..XX.X..X..X.X...X.
XXX.XX...XX..X.X....X....X..X.X.....
...X.....X...X....XX....XX..XX..X...
XX..X..XX.......XX..X.X..X...X.X...X
X..X..X..XX..X..XX....X..X..X.X..X..
....X.XX.XX.XX..X...X.X..X..X.X.X...
.X.XX....X..X......X.X...X.X.XX...X.
.......XXX...X..XX....X..X..XX..X.X.
.XX.X...X.......X..X..X..X..XX......
XX...X.X.XX.X...X..XXX...X.X..XXX...
.X..X.....X..X.....XX....X...XX..XX.
....X.X..XX.XX...........X....XX....
XX.X....X....X..XX.XX.X..X...XX..X.X
XX.....X.X......X.....X..X.....XX.X.
....XX...XX..X...XX.X...X...X...X.X.
X....X....X.....XX..........

If you copy all 1000 characters into Microsoft WordPad, make it all a
fixed-pitch font, delete the carriage return/line feed at the end of each
line to combine it all into one long line, and select View > Options... >
Wrap to Window, then various patterns may seem to emerge as you drag the
right edge of the window to widen and narrow it....

Okay, I figure that's considered a barbaric approach.  :-)  Are there any
recommended tools/methods for detecting such patterns?

Thanks for bearing with another newbie email,

-- Jon