A293982 -id:A293982 - cp's OEIS frontend

A294643 Length (= size) of the orbit of n under the "3x+1" map A006370: x -> x/2 if even, 3x+1 if odd. a(n) = -1 in case the orbit would be infinite.

1, 3, 3, 8, 3, 6, 9, 17, 4, 20, 7, 15, 10, 10, 18, 18, 5, 13, 21, 21, 8, 8, 16, 16, 11, 24, 11, 112, 19, 19, 19, 107, 6, 27, 14, 14, 22, 22, 22, 35, 9, 110, 9, 30, 17, 17, 17, 105, 12, 25, 25, 25, 12, 12, 113, 113, 20, 33, 20, 33, 20, 20, 108, 108, 7, 28, 28

Offset: 0

M. F. Hasler, Nov 05 2017

nonn

The orbit of x under f is O(x; f) = { f^k(x); k = 0, 1, 2, ... }, i.e., the set of all points in the trajectory of x under iterations of f.

The famous "3x+1 problem" or Collatz conjecture (also attributed to other names) states that for f = A006370, the trajectory (f^k(x); k >= 0) always ends in the cycle 1 -> 4 -> 2 -> 1, for any integer starting value x >= 0.

			a(0) = 1 = # { 0 }, since 0 -> 0 -> 0 ... under A006370.
a(1) = 3 = # { 1, 4, 2 }, since 1 -> (3*1 + 1 =) 4 -> 2 -> 1 -> 4 etc. under A006370.
a(3) = 8 = # { 3, 10, 5, 16, 8, 4, 2, 1 }, since 3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1 -> 4 etc. under A006370.

Cf. A006370 (Collatz or 3x+1 map), A008908 (number of steps to reach 1), A174221 (the "PrimeLatz" map: add 3 next primes), A293980, A293975 (variant: add the next prime), A293982.

A294659 Largest number in the orbit of n under iteration of the map A293975: x -> x/2 if even, x + nextprime(x) else.

Original entry on oeis.org

0, 8, 8, 8, 12, 8, 20, 8, 20, 12, 24, 12, 32, 20, 32, 16, 36, 20, 44, 20, 44, 24, 52, 24, 56, 32, 56, 28, 60, 32, 68, 32, 72, 36, 72, 36, 80, 44, 80, 40, 84, 44, 92, 44, 92, 52, 100, 48, 104, 56, 104, 52, 112, 56, 116, 56, 116, 60, 120, 60, 128, 68, 132, 64, 132, 72, 140, 68, 140, 72, 144, 72, 152, 80

Offset: 0

M. F. Hasler, Nov 06 2017

nonn

The trajectory under iterations of A293975 seems to end in the cycle 1 -> 3 -> 8 -> 4 -> 2 -> 1, for any positive starting value n. Therefore a(n) >= 8 for all n > 0.
Obviously also a(n) >= n for all numbers, with equality for powers of two 2^k with k >= 3; a(n) >= n + nextprime(n) >= 2n+2 for all odd numbers.
Record values not of the form f(n) = n + nextprime(n) occur for a(1) = 8, a(5) = 12, a(7) = 20, a(11) = 24, a(13) = 32, a(17) = 36, a(19) = 44, a(23) = 52, a(25) = 56, a(29) = 60, a(31) = 68, a(33) = 72, a(37) = 80, a(41) = 84, a(43) = 92, a(47) = 100, ... We see that in most cases, this equals f(f(n)/2). Exceptions are n = 1, 7, 13, 19, 37, 43, 67, 79, 89, 97, ...

Cf. A293975, A293982 (size of the orbit).

A294659(n,S=[n])={while(#S<#S=setunion(S,[n=A293975(n)]),); vecmax(S)}

cp's OEIS Frontend

A294643 Length (= size) of the orbit of n under the "3x+1" map A006370: x -> x/2 if even, 3x+1 if odd. a(n) = -1 in case the orbit would be infinite.

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