A266552 Irregular triangle read by rows giving the 3p+1 sequence of n.

2, 3, 10, 5, 16, 8, 4, 2, 4, 2, 5, 16, 8, 4, 2, 6, 3, 10, 5, 16, 8, 4, 2, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 8, 4, 2, 9, 3, 10, 5, 16, 8, 4, 2, 10, 5, 16, 8, 4, 2, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2

Offset

2,1

Comments

The n-th row of the triangle provides the 3p+1 sequence for n, such that the sequence terminates at the first occurrence of 2. The 3p+1 sequence is a variation of the 3x+1 (Collatz) sequence.

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..71.

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

Example

The irregular array a(n,k) starts:
n\k   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17
2:    2
3:    3  10   5  16   8   4   2
4:    4   2
5:    5  16   8   4   2
6:    6   3  10   5  16   8   4   2
7:    7  22  11  34  17  52  26  13  40  20  10   5  16   8   4   2
8:    8   4   2
9:    9   3  10   5  16   8   4   2
10:   5  16   8   4   2
11:  11  34  17  52  26  13  40  20  10   5  16   8   4   2

Prog

(PARI) row(n) = {print1 (n, ", "); while (n!=2, nn = if (isprime(n), 3*n+1, n/factor(n)[1, 1]); print1(nn, ", "); n=nn); } \\ Michel Marcus, Jan 02 2016
(Python)
from sympy import isprime, primefactors
def a(n):
    if n==2: return [2]
    l=[n, ]
    while True:
        if isprime(n): n = 3*n + 1
        else: n/=min(primefactors(n))
        l+=[n, ]
        if n==2: break
    return l
for n in range(2, 21): print a(n) # Indranil Ghosh, Apr 22 2017

Crossrefs

Cf. A266551 (image of n under the 3p+1 map).

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

Cf. A070165 (irregular triangle read by rows giving trajectory of n in Collatz problem).

Sequence in context: A139693 A182076 A347271 * A263716 A344457 A175899

Adjacent sequences: A266549 A266550 A266551 * A266553 A266554 A266555

Keyword

nonn,tabf

Author

Robert C. Lyons, Dec 31 2015

Status

approved