31.12.16

2016 Sequences

I came up with 7 different sequences in 2016 and I thought I'd put them all in one place as some of them are slightly related to one another.

1. The first sequence is the sequence generated by powers of 2 mod n when n is the product of 2 distinct safe primes. The sequence was published in the OEIS.org with sequence number is A269453. Here are the first few terms of the sequence:

12, 20, 30, 44, 33, 92, 110, 116, 69, 174, 164, 230, 212, 246, 290, 318, 332, 356, 410, 253, 452, 249, 530, 534, 524, 638, 678, 692, 716, 830, 393, 902, 764, 890, 932, 956, 1038, 1166, 1130, 537, 1004, 573, 1334, 1124, 1310, 1172, 1398, 717, 753, 1436, 1730, 913, 1886, 1686, 1790

2. Then came the the sequence of the products of two distinct safe primes, which is now published in the OEIS.org under sequence number A269452

24, 40, 60, 88, 132, 184, 220, 232, 276, 348, 328, 460, 424, 492, 580, 636, 664, 712, 820, 1012, 904, 996, 1060, 1068, 1048, 1276, 1356, 1384, 1432, 1660, 1572, 1804, 1528, 1780, 1864, 1912, 2076, 2332, 2260, 2148, 2008, 2292, 2668, 2248, 2620, 2344, 2796, 2868, 3012, 2872, 3460, 3652, 3772, 3372


3. Looking at these 2 sequences made me discover a third sequence, which I dedicated to my great grandmother Yona because at the time I thought it was the most important result I'd come up with. This sequence consists of safe primes not congruent to -1 mod 8 and it is now published with a sequence number A269454.

5, 11, 59, 83, 107, 179, 227, 347, 467, 563, 587, 1019, 1187, 1283, 1307, 1523, 1619, 1907, 2027, 2099, 2459, 2579, 2819, 2963, 3203, 3467, 3779, 3803, 3947, 4139, 4259, 4283, 4547, 4787, 5099, 5387, 5483, 5507, 5939, 6659, 6779, 6827, 6899, 7187, 7523


The next 3 sequences I came up with involve the Collatz conjecture, which is an open problem in Mathematics. 2 of these sequence were published and the last one was denied and so it remains unpublished because it seemed "artificial"

4. The first sequence of the Collatz trio is the sequence of Collatz primes(A276260), and it only has 9 terms. If a 10th term exists, it would have to be really really large.

5, 13, 17, 53, 61, 107, 251, 283, 1367

5. The second sequence is the sequence of Collatz products, and it has many more terms. It is published under the sequence number A276290 .

25, 35, 55, 65, 77, 85, 95, 115, 133, 143, 145, 155, 161, 185, 203, 205, 209, 215, 217, 235, 253, 259, 265, 287, 295, 305, 329, 341, 355, 365, 371, 391, 395, 403, 407, 415, 427, 437, 445


6. The third sequence of the Collatz trio was never published and here I explain it in some detail.  The first few terms are:

5,27,21,107,85,427,341,1707,1365,6827, 5461

7. The last sequence that I came up with in 2016 is the sequence I described here. I haven't attempted to publish this sequence anywhere yet. The first few terms are:

15, 21, 51, 93, 195, 381, 771, 1533, 3075, 6141

2016 has been the year of discovery for me. I discovered many different results, some of which I even managed to prove. Also in 2016 I learned not to be intimidated by sharing my results with the public.

30.12.16

Just another sequence

In my previous post I talked about the order of 2 mod n where n is an integer such that either n = 2^i + (2^(i-1) - 3) when i is even or n = 2^i + (2^(i-1) + 3) when i is odd.

I conjectured that the order of 2 mod n for such n is i + (i - 2) regardless of whether i is even or odd.

The recurrence relation for generating all such n up to a limit is :

2i + (2(i-1) - 3) if i is even
or 2i + (2i-1 + 3) if i is odd

For the fun of it, I wrote a simple script to generate the first few such integers up to some limit.

/* -------------------------------------------------  
 This content is released under the GNU License  
 http://www.gnu.org/copyleft/gpl.html 
 Author: Marina Ibrishimova 
 Version: 1.0
 Purpose: Generate the 3sequence  
  
 ---------------------------------------------------- */  
function gen_seq(i, limit, a)
{
 var j = i - 1;
 var n = Math.pow(2,j);
 var m = Math.pow(2,i);
 if (i < limit)
 {
  if (i % 2 != 0)
  {
   n = n + 3;
  }
  else
  {
   n = n - 3;
  }
  a.push(m+n);
  i = i + 1;
  gen_seq(i, limit, a)  
 }
 return a;
}

Here's the output for limit = 12:

15, 21, 51, 93, 195, 381, 771, 1533, 3075, 6141

Obviously, there is another way to generate this sequence than the one I use in my script. Each term in the sequence can also be generated by multiplying 3 and 2^i - 1 if i is odd or 3 and 2^i + 1 if i is even.

3*5 = 15 where 5 = 2^2 + 1
3*7 = 21 where 7 = 2^3 - 1
3*17 = 51 where 17 = 2^4 + 1
3*31 = 93 where 31 = 2^5 - 1

and so on.

28.12.16

Between 2 i's

Previously I talked about the order of 2 mod n when n is either equal to 2^i + 1 or 2^i - 1

How about the order of 2 mod n for other integers n, which are between 2^i and 2^(i+1)?