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A247583 Primes extracted from a pseudo-Collatz cycle '3*n-1' by consecutive arithmetic derivatives, here with starting point prime(99147) = 1287511. 1

%I

%S 1287511,1448449,2172673,37122139,44596859,91644073,28996757,3440533,

%T 3870599,4354423,3265817,7348087,8266597,9299921,20924821,31387231,

%U 17655317,19862231,22345009,33517513,50276269,75414403,21499669,34438309,55163509,9817919

%N Primes extracted from a pseudo-Collatz cycle '3*n-1' by consecutive arithmetic derivatives, here with starting point prime(99147) = 1287511.

%C 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.

%C 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).

%C The selection of the prime numbers can be made under the conditions:

%C (1) If AD(i+1,C(k)) = 1 then AD(i,C(k)) is prime.

%C (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.

%C (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.

%C If any of the conditions 1 - 3 are not satisfied then the search for primes by AD in that sequence is hopeless.

%C 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.

%C Table 1

%C i 0 1 2 3 4 5 6 7 8 9 10 ...

%C np 65 33 27 19 10 10 1 3 4 2 0 ...

%C n 65 98 125 144 154 164 165 168 172 174

%C a(n) 17 19 103 71 5 7 101 271 967721 5

%H Freimut Marschner, <a href="/A247583/b247583.txt">Table of n, a(n) for n = 1..174</a>

%e Example for starting point prime(7) = 17. This pseudo-Collatz cycle is repetitive (see A246007).

%e Table 2

%e Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

%e Sequence 17 50 25 74 37 110 55 164 82 41 122 61 182 91 272 136 68 34 17

%e Primes( AD) 17 37 41 61 17 43 131 19 7

%e Table 3

%e i 0 1 2 3 ...

%e np 5 3 1 0 ...

%e n 5 8 9

%e a(n) 17 19 7

%Y Cf. A246007 (length of pseudo-Collatz cycles '3*n - 1' of prime numbers).

%K sign

%O 1,1

%A _Freimut Marschner_, Sep 21 2014

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Last modified July 1 04:11 EDT 2022. Contains 354949 sequences. (Running on oeis4.)