
COMMENTS

This is the "semiFibonacci sequence". The distinct numbers that appear are called "semiFibonacci numbers", and are given in A030068.
a(2n+1) >= a(2n1) + 1 is monotonically increasing. a(2n)/n can be arbitrarily small, as a(2^n) = 1. There are probably an infinite number of primes in the sequence.  Jonathan Vos Post, Mar 28 2006
From Robert G. Wilson v, Jan 17 2014: (Start)
Positions where k occurs:
1: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, ..., = A000079;
2: 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, ..., = 3*A000079 = A007283;
3: 5, 10, 20, 40, 80, 160, 320, 640, 1280, 2560, 5120, 10240, ..., = 5*A000079 = A020714;
4: none in the first 10^6 terms;
5: 7, 14, 28, 56, 112, 224, 448, 896, 1792, 3584, 7168, 14336, ..., = 7*A000079 = A005009;
6: 9, 18, 36, 72, 144, 288, 576, 1152, 2304, 4608, 9216, 18432, ..., = 9*A000079 = A005010;
7: none in the first 10^6 terms;
8: none in the first 10^6 terms;
9: 11, 22, 44, 88, 176, 352, 704, 1408, 2816, 5632, 11264, 22528, ..., = 11*A000079 = A005015;
10: none in the first 10^6 terms;
11: 13, 26, 52, 104, 208, 416, 832, 1664, 3328, 6656, 13312, 26624, ..., = 13*A000079 = A005029;
12: none in the first 10^6 terms;
Values that A030067 takes: 1, 2, 3, 5, 6, 9, 11, 16, 17, 23, 26, 35, 37, 48, ..., = A030068.
(End)
Any integer N which occurs in this sequence first occurs as an oddindexed term a(2k1) = A030068(k1), and thereafter at indices (2k1)*2^j, j=1,2,3,... (Both of these statements follow immediately from the definition of evenindexed terms.) No N can occur a second time as an oddindexed term: This follows from the definition of these terms, a(2n+1) = a(2n) + a(2n1) = a(2n1) + a(n), which shows that the subsequence of oddindexed terms (A030068) is strictly increasing, and therefore equal to the range (or: set) of the semiFibonacci numbers.  M. F. Hasler, Mar 24 2017
The lines in the logarithmic scatterplot of the sequence corresponds to sets of indices with the same 2adic valuation.  Rémy Sigrist, Nov 27 2017
Define the partition subsum polynomial of an integer partition m of n where m = (m_1, m_2, ...m_k) by ps(m,x) = Product_{i=1..k} (1+x^m_i). Expanding ps(m,x) gives 1+a_1 x+a_2 x^2+...+a_n x^n, where a_j is the number of ways to form the subsum j from the parts of m. Then the number of partitions m of n for which ps(m,x) has no repeated root is a(n).  George Beck, Nov 07 2018
