A372450 a(n) = k, if A086893(k) is the first term of A086893 reached on the trajectory of reduced Collatz-function R, when starting from 2n-1, or -1 if no term of A086893 is ever encountered.

1, 2, 3, 4, 4, 4, 4, 6, 4, 4, 5, 6, 4, 6, 4, 6, 4, 6, 4, 4, 6, 4, 4, 6, 4, 4, 6, 6, 4, 4, 6, 6, 4, 4, 4, 6, 6, 7, 4, 4, 6, 6, 7, 4, 4, 6, 6, 6, 6, 4, 4, 6, 4, 6, 6, 6, 7, 4, 4, 4, 6, 4, 6, 4, 6, 4, 4, 4, 6, 4, 6, 6, 6, 6, 4, 9, 4, 6, 4, 6, 6, 6, 6, 6, 4, 6, 4, 6, 4, 4, 4, 6, 4, 4, 6, 4, 6, 6, 4, 6, 9, 4, 4, 6, 4, 4, 8, 6

Offset: 1

Author

Antti Karttunen, May 03 2024

Keywords

• nonn

Comments

The length of the binary expansion of the first term of A086893 that comes along when starting from x = 2*n-1 and then repeating the operation x -> A000265(3*x+1). If 2n-1 itself is in A086893, then its binary length is used.

Terms A016789(n) = 2, 5, 8, 11, 14, 17, ... occur only once in this sequence because A086893(A016789(n)) are all multiples of 3: 3, 21, 213, 1365, 13653, 87381, 873813, 5592405, 55924053, 357913941, ..., while the terms of A075677 never are. Note that all terms > 1 of A086893 are just one or two invocations of R away from 1.

Examples

			a(11) = 5 because the first term of A086893 that occurs on the trajectory of 21 (= 2*11-1) is 21 = A086893(5).
a(14) = 6 because the first term of A086893 that occurs on the trajectory of 27 (= 2*14-1) is A372443(39) = 53 = A086893(6).
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..100000
• Index entries for sequences related to 3x+1 (or Collatz) problem

Crossrefs

Cf. A000265, A000523, A016789, A075677, A086893, A372358, A372443.

Programs

PARI

A000265(n) = (n>>valuation(n,2));
A000523(n) = logint(n,2);
A086893(n) = (if(n%2, 2^(n+1), 2^(n+1)+2^(n-1))\3);
A372358(n) = bitxor(n, A086893(1+A000523(n)));
A372450(n) = { n=2*n-1; while(A372358(n)>0, n=A000265(3*n+1)); (1+A000523(n)); };