A similar conjecture to Fortunate numbers based on sum of the divisors of n

The last months of 2015 I have been trying to get more expressions similar to the ones I found in my previous post "About Fortunate numbers and other similar expressions". So this is a follow-up note with some interesting results. This post is also a mirror of the question I did at MSE (click here).

The generalized idea is finding an expression E(n)$$, whose distances between the value of the expression and the next prime closer to the expression, N(E(n))$$, and from the expression to the previous closer prime, P(E(n))$$, are both prime numbers or 1.

 

I tried several combinations, but the following one seems to work properly so far and is not related with factorials or primorials, so I believe it would be easier to test:

 "Record values" are defined as the subset of values sigma_1(m) of the set of sum of divisors of a natural number m who are bigger than any previous values of the set, up to that value m, so it provides a strictly increasing set of values of sigma_1(m). Those are exactly the values of OEIS A034885: (1,3,4,7,12,15,18,28,31,39,42... )

These are the first elements of the set E(n)E(n), associated to their respective values of OEIS A034885 (= "m's whose sigma_1(m) is part of the sequence OEIS A034885"). They are the Highly Abundant Numbers (OEIS A002093):

{1,2,3,4,6,8,10,12,16,18,20,24,30,36,42,48...}

The point is that tested up to 10^27 both the distances of those elements E(n)E(n) to the previous and following closer primes (when there is a previous positive prime) is also a prime number or 1.

Here is the PARI/GP code to test up to 10^9:

testlimit = 10^9;n=1;exitval=0;prevsumdiv=0;while((n<testlimit) && (exitval==0),dj=divisors(n);sum_div=0;for(t=1,length(dj),sum_div=sum_div+dj[t]);while(sum_div<=prevsumdiv,sum_div=0;n=n+1;dj=divisors(n);for(t=1,length(dj),sum_div=sum_div+dj[t]));prevsumdiv = sum_div;np=nextprime(n+1);npmn=np-n;pp=precprime(n-1);nmpp=n-pp;if((nmpp==1 || isprime(nmpp)) && (npmn==1 || isprime(npmn)), print("n= ",n," sigma1(n)= ",sum_div," N(n)= ",np," N(n)-n= ", npmn ," P(n)= ",pp," n-P(n)= ", nmpp);,exitval=1;));

The big question is.. why does it happen? as far as I can see, it does not happen for other values of sigma_1(m), only for the m's behind the record values of sigma_1(m) as explained above. I can not imagine which possible reason is behind that property. I wonder if there are similar expressions based on other non-factorial functions. So far I have tried using the totient function and some sums of divisors of even numbers instead of sigma_1(n), all of them unsuccessfully. I did not find any reference to this possible conjecture yet.

Update 2015/12/24: I have been able to reduce the expression. It is possible to use instead of sigma_1(m) a reduced version of the sum of divisors, I will call it sigma_e(m) defined as the sum of the even composite divisors of m not including m itself in case of being even. In other words, it is the sum of divisors not including the sum of the prime numbers that are divisors, odd divisors, 1 and m itself. E(n) is:

So it is not required to use A034885 anymore. Apart from this expression, I tried also using the sum of the prime divisors of m, and other combinations of partial sums of the divisors of m, including 1 and m itself, but those ones did not work. The even divisors seem to be the key.

This is an example including the interval [1,99].

Update 2016/01/04: I was able to test it for the Highly Abundant Numbers up to 10^27. No counterexamples found.

Update 2016/01/05: Amazingly ∀h∈ {Highly Abundant Numbers}, h>3,tested up to 10^27 no counterexamples found, the closest prime number p<h located to a distance d=(h−p)>1 is also always at a prime distance, so d∈P. If the test is not wrong, these would mean that the even Highly Abundant Numbers greater than 2 have always at least a Goldbach pair of primes. h=p+d,p,d∈P.

Update 2016/01/06: Regarding the Highly Abundant Numbers conjecture I found a similar question from some years ago at MSE here so this idea is not new. The good point of my version is that I was able to avoid the use of the HAN replacing them by the records of the sum of the even non-prime divisors. I have made further tests and it is possible to do the same with the records of the sum of the non-prime multiples of 3 divisors, the distance between the numbers is bigger but the property seems to hold as well.