[seqfan] Smallest and Largest Prime Factors of n+T(n)

[Ali Sada]

Hi Everyone,

 

Please see the two sequences below:

1.      a(n) is the smallest primefactor of n+T(n).

2, 5, 3, 2, 2, 3, 5, 2, 2, 5, 7, 2, 2, 7, 3,2, 2, 3, 11, 2, 2, 5, 13, 2, 2, 13, 3, 2, 2, 3, 17, 2, 2, 17, 5, 2, 2, 19, 3, 2,2, 3, 23, 2, 2, 7, 5, 2, 2, 5, 3, 2, 2, 3, 5, 2, 2, 29, 31, 2, 2, 5, 3, 2, 2, 3,5, 2, 2, 5, 37, 2, 2, 7, 3, 2, 2, 3, 41, 2, 2, 5, 43, 2, 2, 43, 3, 2, 2, 3, 7, 2,2, 47, 5, 2, 2, 7, 3, 2

 
The first appearance of any prime number in this sequence (other than 3 and 5) happens in the natural order (5 appears before the first appearance of 7, and 7 appears before the first appearance of 13,etc.)
 

 

2.      a(n) is the largest primefactor of n+T(n).

2, 5, 3, 7, 5, 3, 7, 11, 3, 13, 11, 5, 13, 17,5, 19, 17, 7, 19, 23, 7, 11, 23, 3, 7, 29, 5, 31, 29, 11, 31, 7, 11, 37, 19, 13,37, 41, 13, 43, 41, 7, 43, 47, 5, 23, 47, 17, 13, 53, 17, 13, 53, 19, 29, 59, 19,61, 59, 7, 61, 31, 11, 67, 17, 23, 67, 71, 23, 73, 71, 5, 73, 37, 13, 79, 11, 13,79, 83, 7, 41, 83, 29, 17, 89, 29, 13, 89, 31, 47, 23, 31, 97, 19, 11, 97, 101,17, 103

 

This sequence is more straightforward since a(n)is either the largest prime factor of n or the largest prime factor of n+3. Thatmeans a(p)=p for all primes >2. 

The two sequences have the same values at n=1,2,3, and 6. 
In some values, a1(n)*a2(n)=n+T(n). That happens at n=3,4,7,10,11,14,19,23,26,31,34,38,….It could be a third sequence. 

 

Best,

 

Ali