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A266552 Irregular triangle read by rows giving the 3p+1 sequence of n. 0
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 (list; graph; refs; listen; history; text; internal format)
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: A112417 A139693 A182076 * A263716 A175899 A328613

Adjacent sequences:  A266549 A266550 A266551 * A266553 A266554 A266555

KEYWORD

nonn,tabf

AUTHOR

Robert C. Lyons, Dec 31 2015

STATUS

approved

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Last modified April 16 05:26 EDT 2021. Contains 343030 sequences. (Running on oeis4.)