a(1) = 1 is the smallest available choice at that point and does not lead to a contradiction. It means that after this term, the term at index 1 + 1 = 2 must be visited.
a(2) cannot be a prime since these are >= 2 and would "point" to a nonpositive index. The smallest available choice is a(2) = 4, which means that after this term, the term at index 2 + 4 = 6 must be visited.
a(3) also cannot be prime, because 2 would lead back to 3 - 2 = 1 and give a loop, and other primes are too large to specify a valid step to the left. Thus a(3) = 6 is the smallest possible choice, leading to index 3 + 6 = 9 after this term is visited.
Similarly a(4) = 8, leading to index 4 + 8 = 12 after this term.
Then a(5) can be equal to the smallest available number, 2, leading to index 5 - 2 = 3 after this term is visited.
Therefore a(6) cannot be 3, which would lead to 6 - 3 = 3, but a(3) already has a(5) as predecessor. Larger primes aren't possible either, so the smallest possible choice is a(6) = 9, leading to 6 + 9 = 15.
And so on.
After 1000 terms, the smallest unused number is the prime 787, and the earliest term which does not yet have a predecessor is a(226) = 238.
After 2000 terms, the smallest unused number is the prime 1583, and the earliest term which does not yet have a predecessor is a(420).
After 10^4 terms, the smallest unused number is the prime 8219, and the earliest term which does not yet have a predecessor is a(1784).
It appears that these limits increase roughly linearly, which justifies the conjecture that all numbers will eventually appear and have a predecessor.
The trajectories involving the first few terms are:
a(1)=1 -> a(2)=4 -> a(6)=9 -> a(15)=18 -> a(33)=36 -> a(69)=76 -> a(145)=103 -> a(42)=48 -> a(90)=96 -> a(186)=200 -> a(386)=404 -> a(790)=820 -> a(1610)=1666 -> a(3276)=3369 -> a(6645)=6807 -> ...
... -> a(6461)=5261 -> a(1200)=937 -> a(263)=193 -> a(70)=47 -> a(23)=13 ->
a(10)=5 -> a(5)=2 -> a(3)=6 -> a(9)=12 -> a(21)=25 -> a(46)=51 -> a(97)=67 ->
a(30)=34 -> a(64)=70 -> a(134)=144 -> a(278)=294 -> a(572)=596 ->
a(1168)=1210 -> a(2378)=2451 -> a(4829)=3907 -> a(922)=957 -> a(1879)=1939 ->
a(3818)=3922 -> a(7740)=7917 -> ...
... -> a(4286)=3461 -> a(825)=858 -> a(1683)=1737 -> a(3420)=2741 -> a(679)=708 ->
a(1387)=1087 -> a(300)=223 -> a(77)=53 -> a(24)=27 -> a(51)=56 ->
a(107)=116 -> a(223)=163 -> a(60)=66 -> a(126)=89 -> a(37)=23 -> a(14)=7 ->
a(7)=3 -> a(4)=8 -> a(12)=15 -> a(27)=32 -> a(59)=65 -> a(124)=133 ->
a(257)=273 -> a(530)=552 -> a(1082)=1124 -> a(2206)=2271 -> a(4477)=4593 ->
a(9070)=9275 -> ...
We can modify the sequence from a certain index on, in order connect the trajectories through a(3) = 6 and a(4) = 8 earlier. For example, one variant which has the same terms up to a(10) but connects all these quite early yields (breaking lines before "local minima"):
a(1)=1 -> a(2)=4 -> a(6)=9 -> a(15)=7 -> a(8)=10 -> a(18)=11 ->
a(7)=3 -> a(4)=8 -> a(12)=14 -> a(26)=13 -> a(13)=15 -> a(28)=17 ->
a(11)=16 -> a(27)=18 -> a(45)=31 ->
a(14)=20 -> a(34)=21 -> a(55)=23 -> a(32)=22 -> a(54)=37 -> a(17)=24 ->
a(41)=19 -> a(22)=25 -> a(47)=26 -> a(73)=53 ->
a(20)=28 -> a(48)=29 -> a(19)=27 -> a(46)=30 -> a(76)=47 ->
a(29)=32 -> a(61)=33 -> a(94)=71 -> a(23)=34 -> a(57)=41 ->
a(16)=35 -> a(51)=36 -> a(87)=43 -> a(44)=38 -> a(82)=39 -> a(121)=97 ->
a(24)=40 -> a(64)=42 -> a(106)=73 -> a(33)=44 -> a(77)=67 -> a(10)=5 ->
a(5)=2 -> a(3)=6 -> a(9)=12 -> a(21)=45 -> ...
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