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User: John Laky

John Laky has authored 2 sequences.

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

3, 5, 7, 25, 29, 41

Offset: 1

John Laky, Jul 14 2024

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.

			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.

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.

Cf. A058047.

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

A291352 Numerators of the Collatz parity generator fractions.

Original entry on oeis.org

1, 3, 7, 191, 25983, 2663063519, 7160414743310596607, 208229791652776621349215306687873023871, 45218742589363955614633859184750699518167124046103786511727246869388335095775

Offset: 1

John Laky, Aug 22 2017

nonn

a(n) is a numerator of a fraction of the form a(n)/(2^2^n + 1) whose expansion into a repeating binary string models the n-th parity in the Collatz orbit of an integer.

			a(1)=1 => f(1) = 1/3 => b(1) = 101010...
a(2)=3 => f(2) = 3/5 => b(2) = 10011001...
elements in b(2) determine the parity of elements at index 2 in the Collatz parity sequence of an integer.

Cf. A006666.

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User: John Laky

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

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