A324037 The minimal number of iterations to reach 1 of the modified reduced Collatz function, defined for odd numbers 1 + 2*n in A324036 (assuming the Collatz conjecture).

0, 2, 1, 6, 7, 5, 3, 7, 4, 8, 2, 6, 9, 48, 7, 46, 10, 5, 8, 14, 47, 11, 6, 45, 9, 10, 4, 49, 12, 13, 8, 47, 10, 11, 5, 44, 50, 5, 9, 15, 9, 48, 3, 12, 12, 40, 7, 46, 51, 10, 10, 38, 16, 43, 49, 30, 4, 13, 8, 14, 41, 19, 47, 20, 52, 11, 11, 16, 39, 17, 6

Offset

0,2

Comments

The Collatz conjecture is that a(n) is finite. If 1 should never be reached then a(n) = -1.

Compare this sequence with the analogous one A075680(n+1) for the reduced Collatz map of A075677.

a(n) gives also the minimal number of iterations of the Vaillant-Delarue map f, defined in A324245, acting on n to reach 0 (assuming the Collatz conjecture).

For the link to the Vaillant-Delarue paper (where fs is called f_s) see A324036.

Links

Table of n, a(n) for n=0..70.

Formula

fs^[a(n)](1 + 2*n) = 1 but fs^[a(n)-1](1 + 2*n) is not 1 (for all n with finite a(n)), where fs is the modified reduced Collatz map defined for 1 + 2*n in A324036(n), for n >= 1, and a(0) = 0.

Example

a(4) = 7 because 1 + 2*4 = 9 and the 7 fs iterations acting on 9 are 7, 11, 17, 13, 3, 5, 1.
Compare this to the reduced Collatz map given in A075677 which needs only 6 = A075680(5) iterations 7, 11, 17, 13, 5, 1. The additional step in the fs case follows 13 == 5 mod(8).

Crossrefs

Cf. A075677, A075680(n+1), A324036, A324245.

Sequence in context: A217922 A196554 A244647 * A160348 A047708 A256277

Adjacent sequences: A324034 A324035 A324036 * A324038 A324039 A324040

Keyword

nonn

Author

Nicolas Vaillant, Philippe Delarue, Wolfdieter Lang, May 09 2019

Status

approved