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A362940
Consider the Collatz trajectory from n to 1, assuming the Collatz conjecture is true. Then a(n) = number of terms in the trajectory that are greater than 1 and congruent to 1 mod 3. If n never reaches 1, set a(n) = -1.
1
0, 0, 3, 1, 2, 3, 9, 1, 10, 3, 7, 3, 5, 9, 8, 2, 6, 10, 11, 3, 3, 8, 7, 3, 13, 5, 61, 10, 9, 8, 59, 2, 14, 7, 6, 10, 12, 11, 18, 4, 60, 3, 17, 8, 8, 8, 57, 3, 14, 13, 12, 6, 5, 61, 62, 10, 18, 10, 17, 8, 10, 59, 58, 3, 15, 14, 15, 7, 7, 7, 56, 10, 64, 12, 7, 12
OFFSET
1,3
COMMENTS
The terms in the trajectory counted by a(n) might be called "branch points", since they are exactly the numbers that can be reached in more than one way under the Collatz map. So a(n) is a measure of the "Collatz complexity" of n. The term (with a slightly different definition) was suggested by Emanuel Landeholm.
LINKS
EXAMPLE
The Collatz trajectory of 7 is 7 22 34 17 52 26 13 40 20 10 5 16 8 4 2 1, which contains 9 terms > 1 and 1 mod 3, so a(7) = 9.
MATHEMATICA
Table[Count[NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, k, # > 1 &], _?(And[# != 1, Mod[#, 3] == 1] &)] , {k, 100}] (* Michael De Vlieger, Sep 11 2023 *)
CROSSREFS
Sequence in context: A209859 A354092 A262218 * A331695 A245538 A080890
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Sep 11 2023, suggested by a sequence submitted by Emanuel Landeholm on Sep 10 2023 but later withdrawn, which had a somewhat different definition and contained errors.
STATUS
approved