Examples of integer sequences coincidences

Question

For the time being, the OEIS website contains almost $300000$ sequences. Each of these sequences is the mark of a specific mathematical concept. Sometimes two (or more) distinct concepts have the same mark, which suggests a connection between a priori independent mathematical areas. The most famous example like that is perhaps the Catalan numbers sequence: A000108.

Question: What are the examples of pair of integer sequences coinciding on all the known terms, but for which the coincidence for all the terms is unknown?

Cheating is not allowed. By cheating I mean artificial examples like:
$u_n = v_n =n$ for $n \neq 10$, and if RH is true then $u_{10} = v_{10} = 10$, else $u_{10}+1 = v_{10} = 1$.
The existence of an OEIS entry could act as safety.

EDIT: I would like to point out that all the answers below are about pair of integer sequences which were already conjectured to be the same, and of course they are on-topic (and some of them are very nice). Note that such examples can be found by searching something like "conjectured to be identical" on OEIS, as I did for some of my own examples below...
Now, a more surprising kind of answer would be a (non-cheating) pair of integer sequences which are the same on the known entries, but for which there is no evidence a priori that they are the same for all the entries or that they are related (i.e. the precise meaning of a coincidence). Such examples, also on-topic, could reveal some unexpected connections in mathematics, but could be harder to find...

Answer 1

A historical example, in the sense that the conjectural equality has been refuted: A180632 (Minimum length of a string of letters that contains every permutation of $n$ letters as sub-strings) was conjectured equal to A007489 ($\sum_{k=1}^n k!$).

The exact value of A180632 at $n=6$ is still unknown, but it must be less than the conjectured value of $1!+2!+\cdots+6!=873$, because the following string of length 872 contains every permutation of 123456 as a substring:

12345612345162345126345123645132645136245136425136452136451234651234156234152634152364152346152341652341256341253641253461253416253412653412356412354612354162354126354123654132654312645316243516243156243165243162543162453164253146253142653142563142536142531645231465231456231452631452361452316453216453126435126431526431256432156423154623154263154236154231654231564213564215362415362145362154362153462135462134562134652134625134621536421563421653421635421634521634251634215643251643256143256413256431265432165432615342613542613452613425613426513426153246513246531246351246315246312546321546325146325416325461325463124563214563241563245163245613245631246532146532416532461532641532614532615432651436251436521435621435261435216435214635214365124361524361254361245361243561243651423561423516423514623514263514236514326541362541365241356241352641352461352416352413654213654123

Answer 2

[EDITED] The classic example is A000396: "Perfect numbers n: n is equal to the sum of the proper divisors of n" and A000668(n)*(A000668(n)+1)/2 where A000668 are the Mersenne primes.

They are the same if and only if there are no odd perfect numbers.

See also sequences which agree for a long time.

Answer 3

Just another instance of the (second) Strong Law of Small Numbers:

We have A157656(n) = A059100(n-1) for all known terms (i.e., $n\leq 6$), but it's also known that A157656(29) > A059100(28). So, the two sequences diverge somewhere.

Answer 4

Some sequences have conjectured formulas, for example the following which I encountered recently, A227404.

$a_n$ is the total number of inversions, among all permutations on $[n]$ that is a single cycle in the cycle decomposition. It is conjectured that $a_n = n! (3n-1)/12$, and I find it rather amazing that there is no proof of this.

Answer 5

1. The least prime $p$ such that $p+2n$ is also prime: A020483$(n)$, and the smallest number $x$ such that $\sigma(x+2n) = \sigma(x)+2n$: A054906$(n)$.

2. The smallest prime in which a digit appears $n$ times: A084673$(n)$, and the smallest prime containing exactly $n$ $1$'s: A037055$(n)$, for $n>1$.

3. The number of subwords of length $n$ in the infinite word generated by $a \to aab, \ b \to b$ : A006697$(n)$, and the maximal number of distinct nonempty substrings of any binary string of length $n$, plus one: A094913$(n)+1$.

4. The number of distinct values taken by ${\omega}$^${\omega}$^${\dots}$^${\omega}$ (with $n$ $\omega$'s and parentheses inserted in all possible ways) where $\omega$ is the first transfinite ordinal omega: A199812$(n)$, and the number of unlabeled rooted trees with at most $n$ nodes A087803$(n)$, minus $n$ plus one: A255170$(n)$.

5. The number of transitive permutation groups of degree $n$: A002106$(n)$ is conjectured to be the number of Galois groups for irreducible polynomials (over $\mathbb{Q}$) of order $n$ (such groups are transitive). It is a particular case of the Inverse Galois problem.

Answer 6

Another example I know is A191363, i.e., numbers $n$ such that $\sigma(n) = 2n - 2$. According to OEIS all known terms are $(a_k-1)a_k/2$ where $a_k$ is the sequence of Fermat primes (A019434).

Answer 7

For example Recurrence relations of this type:$b_{n+1}+b_{n-1}=(n\alpha + \beta)b_n, n\geq 1 , b(0) = 0, b(1) = 1.$. with $\alpha, \beta$ real or complex appear in several contexts see sequences :A053983, A053984, A058797, A058798