Twin Prime Centers - Are they unusual numbers?

Twin Prime Centers - Are they unusual numbers?

If p and p+2 are both prime numbers then they are called twin primes or a prime pair. And let's call the number p+1 the center of the twin prime pair.

Let D(n) denote the number of factors (divisors) of the integer n. We don't count 1 and n as divisors, so for any prime p we have D(p)=0. And let DP(n) denote the number of prime factors of n.

We're interested in D(p+1) which is the number of divisors of the twin prime center. Is there anything unusual about this? Let's look at the first few twin primes..

(3,5) and D(4)=1 and DP(4)=1
(5,7) and D(6)=2 and DP(6)=2
(11,13) and D(12)=4 and DP(12)=2
(17,19) and D(18)=4 and DP(18)=2
(29,31) and D(30)=6 and DP(30)=3
(41,43) and D(42)=6 and DP(42)=3
(59,61) and D(60)=10 and DP(60)=3
(71,73) and D(72)=6 and DP(72)=2
(101,103) and D(102)=6 and DP(102)=3
(107,109) and D(108)=10 and DP(108)=2
(137,139) and D(138)=6 and DP(138)=3
(149,151) and D(150)=10 and DP(150)=3
(179,181) and D(180)=16 and DP(180)=3
(191,193) and D(192)=12 and DP(192)=2

Note that D is not an increasing function, it can drop, as we see for example with D(60) and D(72). The same goes for DP.

Add the first 2 D values, 1+2=3 and this is a prime number.
Add the first 3 D values, 1+2+4=7 and this is a prime number.
Add the first 4 D values, 1+2+4+4=11 and this is a prime number.
Add the first 5 D values, 1+2+4+4+6=17 and this is a prime number.
Add the first 6 D values, 1+2+4+4+6+6=23 and this is a prime number.

Alas, this trend does not continue, although it's true often.

Content written and posted by Ken Abbott abbottsystems@gmail.com