I refer to an earlier post a few days ago as an introduction to the problem:
b-numbers are positive integers k with k == 5 (mod 8) such that s(k-2) = k; here, s(1), s(2), s(3), … is the hypothetical integer sequence computed by TM #4 (chaotic) of Heiner Marxen and Buntrock (discovered around 1990).
There are 42 b-numbers in the interval [1, 511], that is b-numbers with 1 through 9 bits. There are 22 b-numbers in the interval [512, 1023], that is numbers with 10 bits.
If y is a 10-bit b-number, then y > 512 in fact, and y-512 is a number in the range [1, 511] with 1 to 9 bits.
It turns out that y-512 is a b-number with 1 to 9 bits, when y is a 10-bit b-number. There are 22 possible 10-bit
b-numbers, and each of them gives us a different 1 to 9-bit b-number when we subtract 512 from it.
There are 42 b-numbers in the interval [1, 511], so if x is one of them, x+512 is a 10-bit number satisfying x == 5 (mod 8). For 22 of those 42, x+512 is also a b-number, and in the remaining 20 cases, x+512 isn’t a b-number.
For the 42 values of x, I’m trying to figure out when
x and x+512 are both b-numbers, contrasted with x being a b-number and x+512 not being a b-number.
The table below has 42 rows, one per b-number in the interval [1,511]. First comes the binary expansion of
x+512, x being a b-number with 1 to 9 bits. Then comes x in decimal, x being a 1 to 9-bit b-number. Then comes x+512, a 10-bit number. Next comes ‘1’ if x+512 is not a b-number, and ‘2’ if x+512 is indeed a 10-bit b-number, which happens in 22 cases.
I looked for a relationship between the ‘2’/’1′ which tells us whether x+512 is a b-number, and x+512 ‘s binary expansion.
Up to now, for this and the 13-bit case, I’ve found no
logical relationship.
The table with 42 rows is copied below:
1 0 0 0 0 0 0 1 0 1 5 517 1
1 0 0 0 0 0 1 1 0 1 13 525 1
1 0 0 0 0 1 0 1 0 1 21 533 1
1 0 0 0 0 1 1 1 0 1 29 541 1
1 0 0 0 1 0 0 1 0 1 37 549 1
1 0 0 0 1 0 1 1 0 1 45 557 1
1 0 0 0 1 1 0 1 0 1 53 565 1
1 0 0 0 1 1 1 1 0 1 61 573 1
1 0 0 1 0 0 1 1 0 1 77 589 1
1 0 0 1 0 1 0 1 0 1 85 597 1
1 0 0 1 0 1 1 1 0 1 93 605 1
1 0 0 1 1 0 0 1 0 1 101 613 1
1 0 0 1 1 0 1 1 0 1 109 621 1
1 0 0 1 1 1 0 1 0 1 117 629 2
1 0 0 1 1 1 1 1 0 1 125 637 1
1 0 1 0 0 1 0 1 0 1 149 661 1
1 0 1 0 1 0 1 1 0 1 173 685 1
1 0 1 0 1 1 0 1 0 1 181 693 2
1 0 1 0 1 1 1 1 0 1 189 701 1
1 0 1 1 0 0 1 1 0 1 205 717 1
1 0 1 1 0 1 0 1 0 1 213 725 2
1 0 1 1 0 1 1 1 0 1 221 733 1
1 0 1 1 1 0 0 1 0 1 229 741 1
1 0 1 1 1 0 1 1 0 1 237 749 2
1 0 1 1 1 1 0 1 0 1 245 757 2
1 0 1 1 1 1 1 1 0 1 253 765 2
1 1 0 0 1 1 0 1 0 1 309 821 2
1 1 0 1 0 1 0 1 0 1 341 853 2
1 1 0 1 1 0 1 1 0 1 365 877 2
1 1 0 1 1 1 0 1 0 1 373 885 2
1 1 0 1 1 1 1 1 0 1 381 893 2
1 1 1 0 0 1 0 1 0 1 405 917 2
1 1 1 0 1 0 1 1 0 1 429 941 2
1 1 1 0 1 1 0 1 0 1 437 949 2
1 1 1 0 1 1 1 1 0 1 445 957 2
1 1 1 1 0 0 1 1 0 1 461 973 2
1 1 1 1 0 1 0 1 0 1 469 981 2
1 1 1 1 0 1 1 1 0 1 477 989 2
1 1 1 1 1 0 0 1 0 1 485 997 2
1 1 1 1 1 0 1 1 0 1 493 1005 2
1 1 1 1 1 1 0 1 0 1 501 1013 2
1 1 1 1 1 1 1 1 0 1 509 1021 2
