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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
1287511, 1448449, 2172673, 37122139, 44596859, 91644073, 28996757, 3440533, 3870599, 4354423, 3265817, 7348087, 8266597, 9299921, 20924821, 31387231, 17655317, 19862231, 22345009, 33517513, 50276269, 75414403, 21499669, 34438309, 55163509, 9817919 (list; graph; refs; listen; history; text; internal format)
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

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

Sequence in context: A205663 A205952 A205463 * A206749 A244563 A163681

Adjacent sequences:  A247580 A247581 A247582 * A247584 A247585 A247586

KEYWORD

sign

AUTHOR

Freimut Marschner, Sep 21 2014

STATUS

approved

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Last modified May 25 13:19 EDT 2022. Contains 354071 sequences. (Running on oeis4.)