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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.

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

N. J. A. Sloane, Sep 11 2023, suggested by a sequence submitted by Emanuel Landeholm on Sep 10 2023 but later withdrawn, which had a somewhat different definition and contained errors

nonn

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.

			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.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A006577, A008908.

Table[Count[NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, k, # > 1 &], ?(And[# != 1, Mod[#, 3] == 1] &)] , {k, 100}] (* _Michael De Vlieger, Sep 11 2023 *)

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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.

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