Prime chains: p -> (p + (-1)^{(p-1)/2)} / 2

[Tomasz Ordowski]

Hello Everyone!

Prime chains generated by the map

a(n+1) = (a(n) + (-1)^{(a(n)-1)/2)} / 2, with a(1) = p odd prime.

Equivalently:

• if a(n) == 1 (mod 4), then a(n+1) = (a(n)+1)/2

• if a(n) == 3 (mod 4), then a(n+1) = (a(n)-1)/2

Iterate until the first composite appears and count only the prime terms.

Examples: 47 -> 23 -> 11 -> 5 -> 3 -> 1 (5 primes);

2879 -> 1439 -> 719 -> 359 -> 179 -> 89 -> 45 (6 primes). 

The smallest primes starting chains of length k seem to begin

3, 5, 11, 23, 47, 2879, ...

Question:
Can anyone find longer prime chains for this iteration, and the smallest primes producing chains of length 7, 8, ... ?

(Backward step: a = 2b ± 1.)

Best, 

Tom Ordo 

[Tomasz Ordowski]

Prime chains: 

p -> (p + (-1)^{(p-1)/2}) / 2

Correction of brackets, it should be:

a(n+1) = (a(n) + (-1)^{(a(n)-1)/2}) / 2, with a(1) = p odd prime. 

[Amiram Eldar]

The least prime that generates a chain of 7 is 1065601:
1065601 -> 532801 -> 266401 -> 133201 -> 66601 -> 33301  -> 16651

[Tomasz Ordowski]

Ami, thanks! 

Cf. https://oeis.org/A110092

and https://oeis.org/A110093

(see the last three terms there).

Tom  

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[Amiram Eldar]

and the least prime that generates a chain of 8 is 1985902081:
1985902081 -> 992951041 -> 496475521 -> 248237761 -> 124118881 -> 62059441 -> 31029721 -> 15514861

[Tomasz Ordowski]

Very nice!

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[Tomasz Ordowski]

Cf. https://oeis.org/A110059

See a(8). 

[Tomasz Ordowski]

The largest prime that gives a full prime chain (ending in 3) is p = 47.

There will be no such restriction if we change the definition of the prime chain as follows.

Namely p -> Odd(p - (-1)^{(p-1)/2}), where Odd(m) is the odd part of m (its greatest odd divisor). 

Backward iteration (p -> p2^n +- 1 for some n > 1) allows us to create arbitrarily long prime chains (starting from 1). 

In the first step, we obtain the Fermat(+) or Mersenne(-) primes. In the next steps, the tree will have numerous prime branches. 

[Dave Consiglio]

a(9) =  21981381119

My computer has been chewing on it all weekend - a(10) > 174872140559

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