A374670 Generalized Collatz sequences: coefficients resulting in a cycle containing 1.

3, 5, 7, 25, 29, 41

Offset: 1

Author

John Laky, Jul 14 2024

Keywords

• nonn
• more

Comments

For each integer coefficient C >= 3, check the generalized Collatz sequence of each integer N > 1 defined by c(1) = N, c(n+1) = c(n) * C + 1 if F > C, otherwise c(n+1) = c(n) / F, where F is the smallest factor of c(n). If all integers reduce to 1, C belongs in the sequence.

A058047 is the primes in this sequence, as proposed by Zhongfu and Shiming; this sequence extends to consider composite C.

All terms are as yet only conjectures. Link includes github of code implementing CPU and GPU tests of these coefficients. Tests include reductions of integers up to 10^7 and random integers of size 2^2048.

Examples

			The first term in the sequence, 3, is the normal Collatz formula C_3 is defined as:
  n / 2 if n is congruent to 0 mod 2,
  otherwise, 3*n+1.
Note that 3 is the first element of this sequence iff the Collatz Conjecture is true.
The next term, 5, creates C_5 is defined as:
  n / 2 if n is congruent to 0 mod 2,
  n / 3 if n is congruent to 0 mod 3,
  otherwise, 5*n+1.
However, the coefficient 11 is not in this sequence, as it creates the formula C_11:
  n / 2 if n is congruent to 0 mod 2,
  n / 3 if n is congruent to 0 mod 3,
  n / 5 if n is congruent to 0 mod 5,
  n / 7 if n is congruent to 0 mod 7,
  otherwise, 11*n+1.
C_11 creates a nontrivial loop 188, 518, 408, 188. The loop is nontrivial because it does not contain 1. Thus, 11 does not belong in the sequence.
		

Links

• J. C. Lagarias, The 3x + 1 Problem: An Annotated Bibliography, Source #195 (1963-1999), p. 73, arXiv:math/0309224 [math.NT], 2003-2011.
• J. C. Lagarias, The 3x + 1 Problem: An Annotated Bibliography, II (2000-2009), arXiv:math/0608208 [math.NT], 2006-2012.
• John Laky, Python Collatz tests.

Crossrefs

Cf. A058047.

Programs

Python

import sympy
BIT_LIMIT = 100000
MAX_TESTS = 10000000
def _reduce(c,n):
    seen = []
    while n != 1:
        for p in sympy.sieve.primerange(c):
            while n % p == 0: n = n // p
        if n==1: break
        n = c*n+1
        if n.bit_length() > BIT_LIMIT or n in seen: break
        seen.append(n)
    return n
def is_A374670(c):
    for i in range(1, MAX_TESTS):
        if _reduce(c, i) != 1: return False
    return True