A368241 -id:A368241 - cp's OEIS frontend

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

Hendrik Kuipers, Dec 16 2023

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.

			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

Cf. A368241.

Cf. A000720, A005171, A010051, A006512.

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

T(n,0) = n.

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

A368690 Number of terms in the trajectory from n to 2 of the map x -> A368241(x), or -1 if n never reaches 2.

Original entry on oeis.org

5, 2, 4, 2, 8, 3, 9, 6, 7, 2, 8, 7, 10, 6, 9, 2, 7, 6, 9, 6, 17, 8, 16, 8, 6, 5, 8, 2, 3, 16, 7, 15, 7, 5, 9, 10, 7, 6, 8, 2, 15, 6, 14, 6, 5, 10, 8, 9, 6, 5, 7, 7, 16, 3, 14, 5, 13, 2, 14, 4, 9, 7, 8, 5, 5, 13, 6, 6, 15, 2, 13, 11, 10, 12, 28, 5, 13, 3, 8, 6, 7, 15, 3, 4, 12, 5

Offset: 4

Hendrik Kuipers, Jan 03 2024

nonn

It is conjectured that every starting n reaches 2 eventually.
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.

			For n=4 the trajectory is 4 -> 6 -> 9 -> 13 -> 2 (row 4 of A368196) which has a(4) = 5 terms.

Cf. A368241.

Cf. A000720, A005171, A010051, A006512.

f(n) = if (isprime(n), n - precprime(n-1), n + primepi(n)); \\ A368241
a(n) = my(k=1); while ((n = f(n)) != 2, k++); k+1; \\ Michel Marcus, Jan 03 2024

cp's OEIS Frontend

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).

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