[seqfan] A process connected with proper divisors of positive integers

[Vladimir Shevelev]

Dear SeqFans,
 
Let n be a positive integer. We begin our process with n+1. Let n+1 have proper divisors >1: d_1, ..., d_k. Consider proper divisors of  numbers n+d_1,...,n+d_k which not earlier appeared. Let they d^(1)_1,...,d^(1)_t. Further, consider proper
divisors of  numbers n+d^(1)_1,...,n+d^(1)_t which  not earlier appeared, etc. Let a(n) be the total number of different
divisors which appeared in the considered process. Then sequence {a(n)} begins 0,0,1,0,3,0,5,1,5,0,9,0,11,2,3,0,15,...(A226770).
For example, for n=9,  the proper divisors >1 of n+1 are 2,5; consider n+2=11 and n+5=14. These numbers give only one "new" proper divisor >1 7; the "new" proper divisors >1 of n+7=16 are 4,8 and n+4=13, n+8=17 do not have proper divisors >1. The set of   
proper divisors of all considered sums is {2,5,7,4,8}. It contains 5 elements. Thus 
a(9)=5. 
It is clear that a(n)=0 iff n=p-1, where p is prime. Furthermore, I believe that a(p)=p-2. What one can say about other n's?
 
Best regards,
Vladimir

 Shevelev Vladimir‎