

A220182


Number of changes of parity in the Collatz trajectory of n.


1



0, 1, 4, 1, 2, 5, 10, 1, 12, 3, 8, 5, 4, 11, 10, 1, 6, 13, 12, 3, 2, 9, 8, 5, 14, 5, 82, 11, 10, 11, 78, 1, 16, 7, 6, 13, 12, 13, 22, 3, 80, 3, 18, 9, 8, 9, 76, 5, 14, 15, 14, 5, 4, 83, 82, 11, 20, 11, 20, 11, 10, 79, 78, 1, 16, 17, 16, 7, 6, 7, 74, 13, 84, 13
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OFFSET

1,3


COMMENTS

For n < 10^10, if n <> 27, f(n) is finite, f(n) < 3n + 1. If n = 27 = 3^3, f(n) = 82 = 81 + 1 = 3^4 + 1 = 3n + 1. I conjecture that for any n <> 27, f(n) is finite, f(n) < 3n + 1.  Sergey Pavlov, Jun 02 2019. Note that this conjecture is stronger than the Collatz conjecture.  Andrey Zabolotskiy, Jun 13 2019


REFERENCES

R. K. Guy, Unsolved Problems in Number Theory, E16


LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000


EXAMPLE

For n=5, Collatz trajectory for 5 is: 5,16,8,4,2,1; hence no. of transitions between odd and even parity is a(5)=2; similarly for n=11, Collatz trajectory gives 11,34,17,52,26,13,40,20,10,5,16,8,4,2,1; implies that a(11)=8.


MATHEMATICA

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; parity[n_] := If[OddQ[n], 1, 0]; Table[p = parity /@ Collatz[n]; If[OddQ[n], 2*Total[p]  2, 2*Total[p]  1], {n, 100}] (* T. D. Noe, Feb 24 2013 *)


PROG

(PARI) next_iter(n) = if(n%2==0, return(n/2), return(3*n+1))
parity(n) = n%2
a(n) = my(x=n, par=parity(x), i=0); while(x > 1, x=next_iter(x); if(parity(x)!=par, i++; par=parity(x))); i \\ Felix FrÃ¶hlich, Jun 02 2019


CROSSREFS

Cf. A006577.
Sequence in context: A046573 A006287 A087225 * A076064 A293570 A016685
Adjacent sequences: A220179 A220180 A220181 * A220183 A220184 A220185


KEYWORD

nonn,look


AUTHOR

Jayanta Basu, Feb 20 2013


STATUS

approved



