An Interesting Prime

Being born on the 3rd April 1949, my date of birth is often represented as 3 - 4 - 49. These numbers when concatenated form the prime number 3449. I was reminded of this number because of the factorisation of the number associated with my diurnal age today, 27592.$$27592=2^3\times 3449=8\times 3449$$So today my life can be divided into exactly eight equal parts, each of them 3449 days long which is about 9.44 years. The previous multiple$$7\times 3449=24143$$occurred on May 10th 2015 when I was still working at the Shanghai Singapore International School. The next multiple$$9\times 3449=31041$$ will fall on March 29th 2034, shortly before my 84th birthday (if I make it that far).

3449 forms the initial prime of a Cunningham chain of the first type with length exactly 3 and so: $$\begin{gathered} 3449\to 2\times 3449+1=6889\text{ (prime)} \\ 6889\to 2\times 6889+1=13799\text{ (prime)} \end{gathered}$$Primes with this property form OEIS A059762. Another prime-related property of 3449 qualifies it for membership in OEIS A088483:

A088483: primes $\mathit{\text{p}}$ such that $p^2+p-1$ and $p^2+p+1$ are twin primes.

For 3449, the twin primes are $11899049$ and $11899051$. 

3449 is also a home prime with a homeliness of 3 because:$$\begin{array}{rl}611& =13\times 47\to 1347\\ 1347& =3\times 449\to 3449\end{array}$$Not all primes are home primes of course. Take 613 as an example of a prime that is not a home prime because it cannot be formed by the concatenation of the prime factors of any number (the prime factors need to be concatenated in ascending order).

3449 is also a member of OEIS A153116:

A153116: primes $p$ such that $p^2+12$ and $p^2-12$ are also primes.

Here the two primes are $11895589$ and $11895613$. Additionally:$$\text{period of}\frac{1}{3449}=\frac{3449-1}{8}=431$$This property qualifies 3449 for membership of OEIS A056213:

A056213: primes $p$ for which the period of reciprocal = ${\displaystyle \frac{p-1}{8}}$.

3449 is a Sophie Germain prime because:$$2\times 3449+1=6899\text{ is prime}$$3449 also features in so-called "Golden Semiprimes" and this qualifies it for membership in OEIS A108544:

A108544: primes that are factors of distinct golden semiprimes (A108540).

I posted about these types of semiprimes in Semiprime Factor Ratios way back on the 26th August 2016. In that post I said that:

A golden semiprime is a number that factors to:

• $p\times q$ where $p$ and $q$ are prime
• $|p\varphi -q|<1$ where $\varphi ={\displaystyle \frac{\sqrt{5}+1}{2}}$

In the case of 3449, it is the $q$ and $p=2131$ and the golden semiprime is:$$7349819=2131\times 3449$$The OEIS mentions 314 sequences in which 3449 makes an appearance and I've only dealt with a few of them here. However, I see 3449 as an important number in my life and didn't want its current occurrence to pass unnoticed.