A266551 Image of n under the 3p+1 map, which is a variation of the 3x+1 (Collatz) map.

7, 10, 2, 16, 3, 22, 4, 3, 5, 34, 6, 40, 7, 5, 8, 52, 9, 58, 10, 7, 11, 70, 12, 5, 13, 9, 14, 88, 15, 94, 16, 11, 17, 7, 18, 112, 19, 13, 20, 124, 21, 130, 22, 15, 23, 142, 24, 7, 25, 17, 26, 160, 27, 11, 28, 19, 29, 178, 30, 184, 31, 21, 32, 13, 33, 202, 34, 23, 35, 214, 36, 220, 37, 25

Offset

2,1

Comments

The 3p+1 map is defined for n >= 2 as follows: if n is prime, then map n to 3n+1; otherwise, map n to n divided by the smallest prime divisor of n.

The 3p+1 sequence for n >= 2 is defined as follows: b(0) = n; b(n+1) = 3 * b(n) + 1 if b(n) is prime; otherwise, b(n+1) = b(n) divided by the smallest prime factor of b(n).

It seems that all 3p+1 sequences reach 2. This has been verified for n up to 5*10^8. Once a 3p+1 sequence reaches 2, it repeats the following cycle: 2, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, ...

Links

Table of n, a(n) for n=2..75.

Jack Brennen, Felice Russo, A Collatz-like sequence.

Example

For n = 2, a(2) = 2*3 + 1 = 7, because 2 is prime.
For n = 6, a(6) = 6 / 2 = 3, because 6 is composite; 2 is the smallest prime factor of 6.

Mathematica

a[n_]:=If[PrimeQ[n], 3n + 1, n/FactorInteger[n][[1, 1]]]; Table[a[n], {n, 2, 100}] (* Indranil Ghosh, Apr 22 2017 *)

Prog

(PARI) a(n) = if (isprime(n), 3*n+1, n/factor(n)[1, 1]); \\ Michel Marcus, Jan 02 2016
(Python)
from sympy import isprime, primefactors
def a(n): return 3*n + 1 if isprime(n) else n//min(primefactors(n))
print([a(n) for n in range(2, 101)]) # Indranil Ghosh, Apr 22 2017

Crossrefs

Cf. A175871 (the repeating cycle starting at 2).

Cf. A006370 (image of n under the 3x+1 map).

Sequence in context: A082705 A122577 A180732 * A070405 A010730 A225694

Adjacent sequences: A266548 A266549 A266550 * A266552 A266553 A266554

Keyword

nonn

Author

Robert C. Lyons, Dec 31 2015

Status

approved