A368196 -id:A368196 - cp's OEIS frontend

A368241 a(n) = n - prevprime(n) if n is prime, n + primepi(n) otherwise; a(2) = 2.

1, 2, 1, 6, 2, 9, 2, 12, 13, 14, 4, 17, 2, 20, 21, 22, 4, 25, 2, 28, 29, 30, 4, 33, 34, 35, 36, 37, 6, 40, 2, 43, 44, 45, 46, 47, 6, 50, 51, 52, 4, 55, 2, 58, 59, 60, 4, 63, 64, 65, 66, 67, 6, 70, 71, 72, 73, 74, 6, 77, 2, 80, 81, 82, 83, 84, 6, 87, 88, 89, 4, 92, 2, 95, 96

Offset: 1

Hendrik Kuipers, Dec 18 2023

prevprime(n) = A151799(n) is the largest prime < n.

a(n) = 2 iff n=2 or n is the larger prime of a twin prime pair (A006512).

Cf. A000720, A005171, A010051, A006512, A095117, A151799.

Cf. A368196 (trajectories).

Table[If[n<3, n, If[PrimeQ[n], n-NextPrime[n,-1], n+PrimePi[n]]],{n,75}] (* James C. McMahon, Dec 19 2023 *)

a(n) = if (isprime(n), n - precprime(n-1), n + primepi(n)); \\ Michel Marcus, Dec 18 2023

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

A368241 a(n) = n - prevprime(n) if n is prime, n + primepi(n) otherwise; a(2) = 2.

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