OFFSET
1,12
COMMENTS
The analog of A398792 for the 5x+1 map (A185452). No convergence assumption is involved: merging at the same step index with equal odd count is a finite event decided by the residue class (Terras's parity argument is valid for every map (q*x+1)/2 with q odd), so the count is well defined even though 5x+1 trajectories are believed to diverge.
The density a(n)/2^n at n = 22 is 0.0044, against 0.409 for the Collatz map (A398792) and 0.107 for 7x+1 (A398795). On a step where both members of a pair have the same parity the difference is multiplied by 1/2 (both even) or by q/2 (both odd), so the expected multiplier is (1+q)/4; this equals 1 only for q = 3, and is larger than 1 for q >= 5. The densities observed for q >= 5 are much smaller than for q = 3, but they are nondecreasing in n and they are not ordered by q: q = 7 gives a larger density than q = 5.
LINKS
EXAMPLE
a(11) = 1: the residue r = 909 is the unique class merging within 11 steps, and n = 11 is the smallest level at which any class merges.
PROG
(Python)
def a(n):
c = 0
for r in range(2**n):
x, y, sx, sy = r, r+1, 0, 0
for _ in range(n):
sx += x & 1; sy += y & 1
x = (5*x+1)//2 if x & 1 else x//2
y = (5*y+1)//2 if y & 1 else y//2
if x == y:
c += sx == sy
break
return c
CROSSREFS
KEYWORD
nonn,more,new
AUTHOR
Omar Said, Aug 09 2026
EXTENSIONS
a(23)-a(29) from Sean A. Irvine, Aug 23 2026
STATUS
approved