# Patterns in A129783?

Jon Schoenfield jonscho at hiwaay.net
Wed May 23 05:40:09 CEST 2007

```> 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
> specific nonsquare r values.

I can't prove it, but I wouldn't be surprised if 2 * 3 + 3 = 3^2 were the
only case where r = 3.

Using the prime products up through Prime(4000) * Prime(4001) =  37813 *
37831, and searching for values of r such that

Prime(i) * Prime(i+1) + r is a square

for each r <= 1000,

I get that 238 values of r are found exactly once, beginning with

3, 14, 19, 21, 23, 26, 29, 30, 34, 35, 38, 43, 44, 46, ...

while 81 values of r are found exactly twice, beginning with

10, 62, 86, 92, 122, 138, 146, 161, 179, 181, 188, ...

and 11 values of r are found exactly three times:

190, 318, 404, 482, 518, 534, 676, 698, 788, 803, 908

One value of r is found exactly 4 times:  983

After the above, it's all squares:

Two values of r are found exactly 5 times:  256 and 400

and only 1 value of r was found at each existing frequency F > 5:

324= 18^2 found 13 times
289 = 17^2 found 14 times
169 = 13^2 found 25 times
196 = 14^2 found 29 times
225 = 15^2 found 51 times
144 = 12^2 found 54 times
100 = 10^2 found 82 times
121 = 11^2 found 83 times
64 = 8^2 found 132 times
49 = 7^2 found 188 times
81 = 9^2 found 198 times
16 = 4^2 found 321 times
25 = 5^2 found 373 times
36 = 6^2 found 410 times
4 = 2^2 found 555 times
1 = 1^2 found 570 times
9 = 3^2 found 880 times

Among all positive values of r <= 1000, there were 350 that were found at
least once, i.e., 650 were not found at all.  (I think a few of those 650
would be found if, say, a million primes were used ... but not many.)

-- Jon

1,2,3,4,5,6,7,9,10,13,14,17,18,19,22,23,25,26,29,30,31,34,38,etc...
(i.e., integers which cannot be expressed as n(n+E), where E is an even
integer greater than zero)

8,12,15,16,20,21,24,27,28,32,33,35,40,44,45,48,etc...
(i.e., integers which can be expressed as n(n+E), where E is an even integer
greater than zero)

1,4,1,9,16,4,1,9,36,4,16,1,9,100,4,1,etc...

I've tried searching using small bits of each of these sequences but haven't
found any matches yet in the OEIS. Thanks!

```