0,3

A375495(k) is the number of ways to reach k by the minimum length construction and the present sequence is those k for which A375495(k) = 1.

This sequence is infinite, since k = f(f(f(...(1)))) (A003462) is always a unique minimum construction.

Experimentally, it is observed that many of the integers with unique smallest constructions have one or more g(x) steps in their construction and the position of those g(x) operations does not follow an obvious pattern; meaning that these unique constructions are complex and may be related to the complexity of the Hailstone sequences (A127933).

9 is a term since the shortest sequence of f and g to reach 9 (length A375494(9) = 5) is unique g(f(g(f(f(1))))) = 9.

Here are some of the unique, smallest constructions. All other positive integers smaller than 28 do not have unique, smallest constructions.

0 = g(1)

1 = 1

2 = g◦f(1)

4 = f(1)

6 = g◦f◦f(1)

7 = f◦g◦f(1)

9 = g◦f◦g◦f◦f(1)

11 = g◦f◦f◦g◦f(1)

13 = f◦f(1)

17 = g◦f◦g◦f◦f◦g◦f(1)

19 = f◦g◦f◦f(1)

20 = g◦f◦f◦f(1)

22 = f◦f◦g◦f(1)

26 = g◦f◦g◦f◦g◦f◦f◦g◦f(1)

28 = f◦g◦f◦g◦f◦f(1)

(Python)

from itertools import product

seq = [None for _ in range(200)]

num = [ 0 for _ in range(len(seq))]

for L in range(0, 23):

for P in product((True, False), repeat=L):

x = 1

for upward in P:

x = 3*x+1 if upward else x//2

if x < len(seq):

if num[x] == 0 or L < seq[x]:

seq[x], num[x] = L, 1

elif L == seq[x]:

num[x] += 1

print(', '.join([str(i) for i, x in enumerate(num) if x==1]))

Cf. A375494 (minimum steps), A375495 (ways attained).

Sequence in context: A278039 A076521 A206910 * A139529 A341291 A307830

Adjacent sequences: A375493 A375494 A375495 * A375497 A375498 A375499

Russell Y. Webb, Aug 31 2024

approved