OFFSET
1,1
COMMENTS
a(n) is defined as a sequence of subsequences of prime numbers extracted from the pseudo-Collatz cycle '3*n-1' , C = c(z) by consecutive arithmetic derivatives AD(i) of C. The starting point here is c(1) = prime(99147) = 1287511, the length is z = 560. The arithmetic derivative AD(i), i >=0 is a tool to select prime numbers out of a given sequence of integers, because the AD of prime numbers is 1.
Let AD(i,C(k)) be the i-th AD of the AD of C(k), then AD(1,C(k)) is the first AD of C(k) with AD(0,C(k)) = C(k). So a(n) = AD(i,C(k)) is a sequence of consecutive values of AD(i) of C(k).
The selection of the prime numbers can be made under the conditions:
(1) If AD(i+1,C(k)) = 1 then AD(i,C(k)) is prime.
(2) If AD(i,C(k)) mod 2 = 1 and AD(i,C(k)) > AD(i+1,C(k)) then AD(i,C(k)) is uneven and is (probably) convergent to a prime number.
(3) If AD(i,C(k)) mod 2 = 0 and AD(i,C(k)) < AD(i+1,C(k)) then AD(i) is even and (probably) divergent.
If any of the conditions 1 - 3 are not satisfied then the search for primes by AD in that sequence is hopeless.
In Tables 1 and 3, i is the number of the AD, np the counting number of primes of the AD and a(n) the last prime number of the i'th AD.
Table 1
i 0 1 2 3 4 5 6 7 8 9 10 ...
np 65 33 27 19 10 10 1 3 4 2 0 ...
n 65 98 125 144 154 164 165 168 172 174
a(n) 17 19 103 71 5 7 101 271 967721 5
LINKS
Freimut Marschner, Table of n, a(n) for n = 1..174
EXAMPLE
Example for starting point prime(7) = 17. This pseudo-Collatz cycle is repetitive (see A246007).
Table 2
Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Sequence 17 50 25 74 37 110 55 164 82 41 122 61 182 91 272 136 68 34 17
Primes( AD) 17 37 41 61 17 43 131 19 7
Table 3
i 0 1 2 3 ...
np 5 3 1 0 ...
n 5 8 9
a(n) 17 19 7
CROSSREFS
KEYWORD
sign
AUTHOR
Freimut Marschner, Sep 21 2014
STATUS
approved