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A377437
Number of multiplication steps for n to reach 1 when iterating 5x+1 removing factors 2 and 3, or -1 if it never reaches 1.
1
0, 0, 0, 0, 4, 0, 1, 0, 0, 4, 2, 0, 3, 1, 4, 0, 2, 0, 1, 4, 1, 2, 6, 0, 2, 3, 0, 1, 5, 4, 4, 0, 2, 2, 3, 0, 5, 1, 3, 4, 3, 1, 1, 2, 4, 6, 7, 0, 4, 2, 2, 3, 7, 0, 7, 1, 1, 5, 6, 4, 3, 4, 1, 0, 4, 2, 2, 2, 6, 3, 7, 0, 4, 5, 2, 1, 5, 3, 3, 4, 0, 3, 4, 1, 8, 1, 5, 2, 6, 4, 2, 6, 4, 7, 16, 0, 1, 4, 2, 2, 5, 2, 2, 3
OFFSET
1,5
COMMENTS
Reaches 1 for all n < 10^9.
A step first removes factors 2 and 3 by the map x->A065330(x), and if this is not 1 sends x->5x+1 (which then counts).
LINKS
EXAMPLE
For n=1, 2, 3, 4, 6,.. (members of A003568) the removal of factors 2 and 3 yields 1, so there is no multiplication step and the entry is 0.
For n=5 the trajectory is 5->26 (13) -> 66 (11) -> 56 (7) -> 36 (1) which needs 4 multiplications.
MAPLE
A377437 := proc(n)
option remember ;
local n6 ;
if A065330(n) = 1 then
0 ;
else
n6 := A065330(n) ;
return 1+procname(1+5*n6) ;
end if;
end proc:
seq(A377437(n), n=1..120) ; # R. J. Mathar, Feb 25 2025
MATHEMATICA
f3[a_] =a/3^IntegerExponent[a, 3]; f23[a_]:=f3[a]/2^IntegerExponent[f3[a], 2]; s={}; Do[m=0; Until[n==1, n=f23[n]; If[n>1, n=5n+1]; m++]; AppendTo[s, m-1], {n, 104}]; s (* James C. McMahon, Feb 25 2025 *)
PROG
(Python)
def A377437(n):
num = 0
while n > 1:
while n % 2 == 0:
n //= 2
while n % 3 == 0:
n //= 3
if n > 1:
n = 5 * n + 1
num += 1
return num
for i in range(1, 50):
print(A377437(i), end=', ')
CROSSREFS
Cf. A065330.
Sequence in context: A115633 A115713 A199571 * A036859 A036861 A120324
KEYWORD
nonn
AUTHOR
Frank A. Stevenson, Oct 27 2024
STATUS
approved