[seqfan] Will all prime numbers appear in these two sequences?

[Ali Sada]

Hi Everyone,
Please seethe two sequences below. I was wondering if it could be proven that all primenumbers appear in any/both of them.  

 

Best,

 

Ali

 

1.      a(1)=1; a(n)= the largestprime factor of a(n-1) if a(n-1) is a composite number >1, otherwisea(n)=a(n-1)+n.

Data: 1, 3, 6, 3, 8, 2, 9, 3, 12, 3, 14, 7, 20, 5,20, 5, 22, 11, 30, 5, 26, 13, 36, 3, 28, 7, 34, 17, 46, 23, 54, 3, 36, 3, 38,19, 56, 7, 46, 23, 64, 2, 45, 5, 50, 5, 52, 13, 62, 31, 82, 41, 94, 47, 102,17, 74, 37, 96, 3, 64, 2, 65, 13, 78, 13, 80, 5, 74, 37, 108, 3, 76, 19, 94,47, 124, 31, 110, 11, 92, 23, 106, 53, 138, 23, 110, 11, 100, 5, 96, 3, 96, 3,98, 7, 104, 13, 112, 7

 

2.      a(1)=1; a(n)= the smallestprime factor of a(n-1) if a(n-1) is a composite number >1, otherwisea(n)=a(n-1)+n.

Data: 1, 3, 6, 2, 7, 13, 20, 2, 11, 21, 3, 15, 3, 17, 32,2, 19, 37, 56, 2, 23, 45, 3, 27, 3, 29, 56, 2, 31, 61, 92, 2, 35, 5, 40, 2, 39,3, 42, 2, 43, 85, 5, 49, 7, 53, 100, 2, 51, 3, 54, 2, 55, 5, 60, 2, 59, 117, 3,63, 3, 65, 5, 69, 3, 69, 3, 71, 140, 2, 73, 145, 5, 79, 154, 2, 79, 157, 236,2, 83, 165, 3, 87, 3, 89, 176, 2, 91, 7, 98, 2, 95, 5, 100, 2, 99, 3, 102