Let $S_k$ be the sequence defined by $a_k(1)=k,\ a_k(n)=a_k(n-1)+(\text{the least prime factor of }a_k(n-1))+1$.
A diagram of these sequences for around $k<100$ is shown below. As you can see, $S_3$ merges with $S_2$ at $211$, and more unexpectedly, $S_{102}$ merges with $S_{50}$ at $64471$. On the other hand, there seem to be some sequences $S_k$ that never merge with $S_l$ for all $l<k$. The $k\leq1000$ which do not seem to merge are: 2, 6, 14, 26, 50, 62, 66, 86, 134, 170, 182, 186, 230, 242, 290, 314, 366, 386, 422, 440, 462, 506, 528, 572, 614, 630, 662, 702, 762, 794, 806, 822, 842, 870, 902, 944, 950, 962. I've checked that each term is distinct for all $a_k(n)<10^{20}$ of these.
My question is, how can we show that $S_2$ and $S_6$, and other pairs never share a term?
[2]─ 5 ┬ 11┬ 23┬ 47┬ 95┬101─203┬211─╶╶╶
8 ┘ 20┘ 44┘ 92┘ 98┘ │
3 ┬ 7 ┬ 15┬ 19┬ 39┬ 43┬ 87┬ 91┬ 99┬103─207┘
4 ┘ 12┘ 16┘ 36┘ 40┘ 84┘ 88┤ 96┘100┘
74─ 77┬ 85┘
78─ 81┤
82┘
[6]─ 9 ┬ 13┬ 27┬ 31┬ 63┬ 67─135─╶╶╶
10┘ 24┘ 28┤ 60┘ 64┘
18─ 21┬ 25┘
22┘
[14]─ 17┬ 35┬ 41┬ 83─167─╶╶(12 terms)╶╶╶2955┬2959─╶╶╶
32┘ 38┘ 80┘ │
30─ 33┬ 37┬ 75┬ 79─159─╶╶(17 terms)╶╶╶2929┘
34┘ 72┘ 76┘
[26]─ 29┬ 59┬119┬127─╶╶╶
56┘116┘124┤
42─ 45┬ 49┬ 57┬ 61─ 123┘
46┘ 54┘ 58┤
48─ 51┬ 55┘
52┘
[50]─ 53┬107─╶╶╶(11 terms)╶╶╶3975┬3979─╶╶╶(17 terms)╶╶╶64465┬64471─╶╶╶
104┘ │ │
110─113─╶╶╶(20 terms)╶╶╶3961┘ │
102─105┬109─219─╶╶╶(38 terms)╶╶╶64467┘
106┘
[62]─ 65┬ 71─143─155─161─169─183─187┬199─╶╶╶
68┘ 90─ 93┬ 97─195┘
94┘
[66]─ 69┬ 73─147┬151─╶╶╶
70┘ │
114─117┬121─133─141─145┘
118┤
108─111┬115┘
112┘
[86]─ 89─179─╶╶╶
edit: I changed the notation to distinguish between terms from different sequences, and I calculated the merging points for $k\leq1000,n\geq5,a_k(n)<10^{20}$: $$\begin{alignat}{3} &a_{42}(7)&&=a_{26}(5)&&=127\\ &a_{108}(8)&&=a_{66}(5)&&=151\\ &a_{90}(5)&&=a_{62}(10)&&=199\\ &a_{3}(12)&&=a_{2}(9)&&=211\\ &a_{240}(6)&&=a_{26}(9)&&=271\\ &a_{284}(6)&&=a_{146}(4)&&=313\\ &a_{150}(6)&&=a_{30}(9)&&=331\\ &a_{336}(9)&&=a_{186}(5)&&=391\\ &a_{210}(7)&&=a_{102}(8)&&=463\\ &a_{510}(8)&&=a_{122}(8)&&=571\\ &a_{122}(8)&&=a_{6}(13)&&=571\\ &a_{602}(5)&&=a_{302}(5)&&=631\\ &a_{302}(5)&&=a_{146}(6)&&=631\\ &a_{326}(7)&&=a_{30}(12)&&=691\\ &a_{362}(9)&&=a_{182}(6)&&=781\\ &a_{390}(5)&&=a_{386}(4)&&=799\\ &a_{426}(5)&&=a_{386}(8)&&=871\\ &a_{434}(5)&&=a_{386}(13)&&=919\\ &a_{890}(5)&&=a_{102}(10)&&=931\\ &a_{236}(5)&&=a_{102}(13)&&=967\\ &a_{942}(5)&&=a_{102}(13)&&=967\\ &a_{966}(6)&&=a_{50}(12)&&=991\\ &a_{254}(6)&&=a_{242}(9)&&=1051\\ &a_{486}(8)&&=a_{242}(9)&&=1051\\ &a_{554}(5)&&=a_{134}(8)&&=1141\\ &a_{270}(10)&&=a_{6}(16)&&=1179\\ &a_{600}(5)&&=a_{596}(5)&&=1219\\ &a_{576}(9)&&=a_{572}(8)&&=1231\\ &a_{596}(6)&&=a_{66}(11)&&=1243\\ &a_{146}(9)&&=a_{66}(14)&&=1275\\ &a_{666}(11)&&=a_{30}(16)&&=1411\\ &a_{710}(7)&&=a_{366}(8)&&=1519\\ &a_{824}(6)&&=a_{194}(10)&&=1681\\ &a_{194}(11)&&=a_{62}(20)&&=1723\\ &a_{846}(6)&&=a_{404}(9)&&=1741\\ &a_{470}(8)&&=a_{230}(11)&&=1981\\ &a_{974}(5)&&=a_{948}(8)&&=1999\\ &a_{590}(5)&&=a_{290}(7)&&=2381\\ &a_{636}(10)&&=a_{66}(20)&&=2611\\ &a_{626}(7)&&=a_{66}(22)&&=2623\\ &a_{30}(25)&&=a_{14}(20)&&=2959\\ &a_{342}(14)&&=a_{14}(21)&&=2971\\ &a_{168}(12)&&=a_{14}(21)&&=2971\\ &a_{752}(7)&&=a_{366}(12)&&=3067\\ &a_{446}(8)&&=a_{110}(11)&&=3763\\ &a_{900}(14)&&=a_{110}(17)&&=3871\\ &a_{110}(24)&&=a_{50}(16)&&=3979\\ &a_{948}(10)&&=a_{50}(17)&&=4003\\ &a_{986}(7)&&=a_{230}(24)&&=4093\\ &a_{992}(15)&&=a_{462}(19)&&=4231\\ &a_{566}(17)&&=a_{290}(11)&&=4789\\ &a_{650}(7)&&=a_{614}(18)&&=5251\\ &a_{656}(9)&&=a_{614}(19)&&=5311\\ &a_{854}(9)&&=a_{62}(26)&&=6931\\ &a_{404}(20)&&=a_{2}(24)&&=7155\\ &a_{740}(13)&&=a_{14}(35)&&=12195\\ &a_{780}(16)&&=a_{762}(17)&&=12739\\ &a_{674}(16)&&=a_{662}(13)&&=21979\\ &a_{770}(32)&&=a_{186}(27)&&=53659\\ &a_{102}(44)&&=a_{50}(35)&&=64471\\ \end{alignat}$$