This new sequence was just published today. I was happy to collaborate with my friend and fellow Illinois Tech alumnus and swimmer, Michael De Vlieger.
A389240
Start the sequence S with S(1) = n and extend S with S(m+1) = S(m)/2 when S(m) is even, otherwise with S(m) + the smallest odd number not yet added. a(n) is the number of steps to reach 1, or -1 if 1 is never reached.
0
0, 1, 3, 2, 8, 4, 4, 3, 6, 9, 9, 5, 33, 5, 5, 4, 9, 7, 7, 10, -1, 10, 10, 6, 7, 34, 34, 6, 13, 6, 6, 5, 34, 10, 10, 8, 9, 8, 8, 11, 10, -1, -1, 11, 11, 11, 11, 7, -1, 8, 8, 35, 34, 35, 35, 7, 8, 14, 14, 7, 96, 7, 7, 6, 14, 35, 35, 11, -1, 11, 11, 9, 35, 10, 10
OFFSET
1,3
COMMENTS
For n = 1, S begins 1, 2, 1, 4, 2. The subsequent terms are A066070.
LINKS
James C. McMahon, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Scatterplot of first 1000000 terms
FORMULA
a(2^k) = k.
a(2^k-1) = k+1 for k > 1.
a(2^k-2) = k+1 for k > 2.
For odd m and a(m) = h, a(2^k*m) = h+k.
EXAMPLE
For n = 5, S begins 5, 6, 3, 6, 3, 8, 4, 2, 1, thus 8 steps to reach 1, so a(5) = 8.
For n = 21, S begins 21, 22, 11, 14, 7, 12, 6, 3, 10, 5, 14, 7, 18, 9. Starting with the 7th step, 3, alternating terms of S are the odd numbers 3, 5, 7, 9..., so the sequence never reaches 1; a(21) = -1.
MATHEMATICA
Table[m = -1;
If[#[[-1]] == 1, -1 + Length[#], -1] &@
If[n < 5,
NestWhileList[If[EvenQ[#], #/2, # + (m += 2)] &, n, # > 1 &],
NestWhileList[If[EvenQ[#], #/2, # + (m += 2)] &, n,
And[#4 > 1, Nand[#1 == #3 - 2, #2 == #4 - 4]] &, 4]], {n, 75}]
CROSSREFS
KEYWORD
sign,new
AUTHOR
James C. McMahon and Michael De Vlieger, Sep 26 2025
STATUS
approved
