# [seqfan] Rank of a prime (Collatz/Syracuse)

Eric Angelini Eric.Angelini at kntv.be
Sun Feb 1 02:01:42 CET 2009

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Hello SeqFans,
the idea behind my question was this one -- I'm trying to produce a
little "number game" like the Collatz-one you all know:
http://mathworld.wolfram.com/CollatzProblem.html

b) decompose it in prime factors (small --> big): "6" = 2.3
c) write under each prime factor it's rank in the primes succession:
2.3
1 2 (because 2 is the 1st prime, and 3 is the 2nd prime)
d) concatenate the ranks to produce a new integer: "12"
e) start again the procedure from there

REMARK: it may happen that the number obtained via concatenation
is itself prime; simply replace it by it's own rank in the primes
succession -- and go on with the procedure from there.

Example for this precise case (with "4"):

a) 4
b) 4 = 2.2
c)     1 1
d) 11 (result of the concatenation), is replaced by 5 (because 11 is the 5th prime)
a) 5 -> replaced by 3
a) 3 -> replaced by 2
a) 2 -> replaced by 1 -- STOP

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I show here an example with 6; a number herunder marked with "quotes" is the result of the concatenation of the ranks of the prime factors written just efore him; this "quoted integer" is decomposed in it's prime factors immediately after (increasing order of the factors, which are separated by dots); then we re-start the full procedure from there:

6 -> 2.3 = "12" -> 2.2.3 = "112" -> 2.2.2.2.7 = "11114" -> 2.5557 = "1733" -> "270" (because 1733 being prime, we replace it with it's rank) -> 2.3.3.3.5 = "12223" -> 17.719 = "7128" -> 2.2.2.3.3.3.3.11 = "11122225" -> 5.5.23.23.29.29 = "33991010" ... etc.

The serie ends on 1 after smthg like 40 jumps, up and down (and a brief encounter with integer "15").

I've tested (by hand), those starts:
1, 2, 3, 4, 5, 6, 7... they all end on 1.

All prime numbers belonging to Neil's sequence A007097 end on 1 (by definition). The composites crossing the integers of A007097 (the same sequence of "Primeth recurrence") all end also on 1 (they are sucked into the vortex!)

Is there an integer not ending on 1 (entering a loop, for instance)?

Best,
E.

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