Suppose that $n=\prod_{i=1}^{k} p_i^{e_i}$ and $m=\prod_{i=1}^{l} q_i^{f_i}$ are prime factorizations of two positive integers $n$ and $m$, with the primes permuted so that $e_1 \le e_2 \cdots \le e_k$, and $f_1 \le f_2 \le \cdots \le f_l$. Then if $k=l$ and $e_i=f_i$ for all $i$, we say that $n$ and $m$ are factorially equivalent. In other words, two integers are factorially equivalent if their prime signatures are identical. In particular, $d(n)=d(m)$ if the two are factorially equivalent.
There's a question I've had for a long time, which is: Are there infinitely many integers $n$ such that $n$ is factorially equivalent to $n+1$? There are numerous curious pairs of consecutive integers for which this holds: $(2,3)$, $(14,15)$, $(21,22)$, $(33,34)$, $(34,35)$, $(38,39)$, $(44,45)$, as well as $(98,99)$, and many more. As you can see, many of them are almost-primes, but the last two pairs are quite striking. Although there are so many of them, a proof that there are infinitely many such pairs seems elusive. Has anyone made any progress on (or even asked) such a question? Does anyone here have a solution or progress for this?
Edit: As an added bonus, the $k$th such $n$, as a function of $k$, seems almost linear. It would be interesting to express and prove an asymptotic formula for this. Can anyone guess heuristically what the slope of this line is?
What I'll add, though I'd like to keep my question focused on the above, is that there are many other questions you can ask: How many integers $n$ are there such that $n$ is factorially equivalent to $n^2+1$, or $n^4+5n+3$, or $2^n+1$? You can generate an almost unending list of seemingly uncrackable number-theoretic conjectures this way.
Many of these questions seem to relate to other well-known number theoretic conjectures. The Twin Prime Conjecture would imply that there are infinitely many $n$ such that $n$ is factorially equivalent to $n+2$. The truth of my question above would imply that there are infinitely many $n$ such that $d(n)=d(n+1)$, a result which has actually been proven, so my conjecture is a strengthening of it. Furthermore, the proof of the infinitude of Mersenne primes would prove the infinitude of $n$ factorially equivalent to $2^n-1$. But beyond all these connections to well-known conjectures, I think the question about and its generalizations are aesthetically interesting.
Best Answer
I'm coming into this late, but am wondering why no one seems to have mentioned the results of Goldston, Graham, Pintz and Yildirim:
http://arxiv.org/pdf/0803.2636.pdf
In particular, their Theorem 4 answers the OPs first question in the affirmative.