Talk:7x±1 problem

15 x ± 1, 15 x ± 3 problem?

     ${\begin{array}{l}\displaystyle {f(n)={\begin{cases}n/2&{\text{if }}n\equiv ~\;\;0{\pmod {2}},\\15n+3&{\text{if }}n\equiv +3{\pmod {8}},\\15n+1&{\text{if }}n\equiv +1{\pmod {8}},\\15n-1&{\text{if }}n\equiv -1{\pmod {8}}.\\15n-3&{\text{if }}n\equiv -3{\pmod {8}}.\\\end{cases}}}\end{array}}$

Interesting or not? — Daniel Forgues 20:28, 2 September 2018 (EDT)

When

n

is congruent to:

• odd:

  n

  is multiplied by about 15 (probability 1/2);
• even:

  n

  is divided by 2 (probability 1/4), divided by 4 (probability 1/8), divided by 8 (probability 1/16), divided by 16 (probability 1/32), ...

Geometric mean: 15^(1/2) * (1/2)^(1/4) * (1/4)^(1/8) * (1/8)^(1/16) * (1/16)^(1/32) * (1/32)^(1/64) is 2.0890113021274 (tends to increase by 108.9% overall..., much longer trajectories, much less chance of falling down to 1...)

Compare with 7x ± 1 problem, when

n

is congruent to:

• odd:

  n

  is multiplied by about 7 (probability 1/2);
• even:

  n

  is divided by 2 (probability 1/4), divided by 4 (probability 1/8), divided by 8 (probability 1/16), divided by 16 (probability 1/32), ...

Geometric mean: 7^(1/2) * (1/2)^(1/4) * (1/4)^(1/8) * (1/8)^(1/16) * (1/16)^(1/32) * (1/32)^(1/64) is 1.4270663974955 (tends to increase by 42.7% overall..., longer trajectories, less chance of falling down to 1...)

Compare with 3x + 1 problem, when

n

is congruent to:

• odd:

  n

  is multiplied by about 3 (probability 1/2);
• even:

  n

  is divided by 2 (probability 1/4), divided by 4 (probability 1/8), divided by 8 (probability 1/16), divided by 16 (probability 1/32), ...

Geometric mean: 3^(1/2) * (1/2)^(1/4) * (1/4)^(1/8) * (1/8)^(1/16) * (1/16)^(1/32) * (1/32)^(1/64) is 0.93423425546443 (tends to decrease by 6.5% overall...)

— Daniel Forgues 23:32, 4 September 2018 (EDT)