A368196 Irregular triangle read by rows where row n is the trajectory starting from n and ending with 2 of the map x -> A368241(x).

4, 6, 9, 13, 2, 5, 2, 6, 9, 13, 2, 7, 2, 8, 12, 17, 4, 6, 9, 13, 2, 9, 13, 2, 10, 14, 20, 28, 37, 6, 9, 13, 2, 11, 4, 6, 9, 13, 2, 12, 17, 4, 6, 9, 13, 2, 13, 2, 14, 20, 28, 37, 6, 9, 13, 2, 15, 21, 29, 6, 9, 13, 2, 16, 22, 30, 40, 52, 67, 6, 9, 13, 2, 17, 4, 6, 9, 13, 2

Offset: 4

Author

Hendrik Kuipers, Dec 16 2023

Keywords

• nonn
• tabf

Comments

It is conjectured that every starting n reaches 2 eventually. (If not then the sequence has an infinite final row.)

Map A368241(x) decreases to the prime gap x-prevprime(x) when x is prime, or increases to x+primepi(x) otherwise, and will reach 2 when x is the greater of a twin prime pair (A006512, preceding prime gap 2).

Prime gaps and x+primepi(x) may become large, but if the twin prime conjecture is true then there would be large twin primes they might reach too.

Examples

			Table T(n,k) begins:
  n\k    0   1   2   3   4   5   6   7   8   9
  --------------------------------------------
   4:    4   6   9  13   2
   5:    5   2
   6:    6   9  13   2
   7:    7   2
   8:    8  12  17   4   6   9  13   2
   9:    9  13   2
  10:   10  14  20  28  37   6   9  13   2
  11:   11   4   6   9  13   2
  12:   12  17   4   6   9  13   2
  13:   13   2
  14:   14  20  28  37   6   9  13   2
  15:   15  21  29   6   9  13   2
  16:   16  22  30  40  52  67   6   9  13   2
  17:   17   4   6   9  13   2
  18:   18  25  34  45  59   6   9  13   2
  19:   19   2
  20:   20  28  37   6   9  13   2
		

Crossrefs

Cf. A368241.

Cf. A000720, A005171, A010051, A006512.

Programs

PARI

row(n) = my(list=List(n)); while(n!=2, n = if (isprime(n), n - precprime(n-1), n + primepi(n)); listput(list, n)); Vec(list); \\ Michel Marcus, Dec 17 2023

Formula

T(n,0) = n.

T(n,k) = A368241(T(n,k-1)) for k >= 1.