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:  $$3449 \rightarrow 2 \times 3449 + 1 = 6889 \text{ (prime)} \\ 6889 \rightarrow 2 \times 6889 +1 = 13799 \text{ (prime)}$$ Primes with this property form OEIS A059762. Another prime-related property of 3449 qualifies it for membership in OEIS A088483:

A088483: primes $\textit{p}$ 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: $$\begin{align} 611 &= 13 \times 47 \rightarrow 1347\\ 1347 &= 3 \times 449 \rightarrow 3449 \end{align}$$ 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 = $\dfrac{p-1}{8}$.

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
• $| \,p \, \phi - q \,| <1$ where $\phi=\dfrac{\sqrt{5}+1}{2}$

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.