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A304715
For any n > 0, if A006666(n) >= 0, then a(n) = Sum_{i = 0..A006666(n)-1} 2^i * [T^i(n) == 0 (mod 2)] (where [] is an Iverson bracket and T^i denotes the i-th iterate of the Collatz function A014682); otherwise a(n) = -1.
2
0, 1, 28, 3, 14, 57, 1896, 7, 7586, 29, 948, 115, 118, 3793, 3824, 15, 474, 15173, 15180, 59, 62, 1897, 1912, 231, 60722, 237, 1102691417057682138372, 7587, 7590, 7649, 137836427132210267296, 31, 242890, 949, 956, 30347, 30350, 30361, 7772616, 119
OFFSET
1,3
COMMENTS
In other words, when a(n) >= 0, the binary representation of a(n) encodes the tripling and halvings steps of the Collatz compressed trajectory of n up to the first occurrence of the number 1 (where zeros and ones respectively denote tripling and halving steps).
FORMULA
a(2^k) = 2^k - 1 for any k >= 0.
a(2*n) = 2*a(n) + 1.
A029837(a(n)+1) = A006666(n).
A000120(a(n)) = A220071(n).
a(A248573(n)) < a(A248573(n+1)) for any n >= 0. - Rémy Sigrist, Nov 09 2018
EXAMPLE
The first terms, alongside the binary representation of a(n) and the Collatz compressed trajectory of a(n) up to the first 1 in reverse order, are:
n a(n) bin(a(n)) rev(traj(n))
-- ---- --------- ------------
1 0 0 (1)
2 1 1 (1, 2)
3 28 11100 (1, 2, 4, 8, 5, 3)
4 3 11 (1, 2, 4)
5 14 1110 (1, 2, 4, 8, 5)
6 57 111001 (1, 2, 4, 8, 5, 3, 6)
7 1896 11101101000 (1, 2, 4, 8, 5, 10, 20, 13, 26, 17, 11, 7)
8 7 111 (1, 2, 4, 8)
9 7586 1110110100010 (1, 2, 4, 8, 5, 10, 20, 13, 26, 17, 11, 7, 14, 9)
10 29 11101 (1, 2, 4, 8, 5, 10)
11 948 1110110100 (1, 2, 4, 8, 5, 10, 20, 13, 26, 17, 11)
12 115 1110011 (1, 2, 4, 8, 5, 3, 6, 12)
13 118 1110110 (1, 2, 4, 8, 5, 10, 20, 13)
14 3793 111011010001 (1, 2, 4, 8, 5, 10, 20, 13, 26, 17, 11, 7, 14)
15 3824 111011110000 (1, 2, 4, 8, 5, 10, 20, 40, 80, 53, 35, 23, 15)
16 15 1111 (1, 2, 4, 8, 16)
17 474 111011010 (1, 2, 4, 8, 5, 10, 20, 13, 26, 17)
18 15173 11101101000101 (1, 2, 4, 8, 5, 10, 20, 13, 26, 17, 11, 7, 14, 9, 18)
PROG
(PARI) a(n) = my (v=0); for (k=0, oo, if (n==1, return (v), n%2, n = (3*n+1)/2, n = n/2; v += 2^k))
CROSSREFS
KEYWORD
nonn,base,look
AUTHOR
Rémy Sigrist, May 17 2018
STATUS
approved