Link OEIS sequences for problems 308, 829, 929

[hjyuh]

Title: Link OEIS sequences for problems 308, 829, 929

Same pattern as #388: adds OEIS identifiers to three entries in data/problems.yaml that currently carry only "possible", keeping the "possible" flag. Each link is to a sequence that is exactly the object in the problem statement (or its inverse), not a related one, and each was checked by independent computation.

Problem 308 -- smallest integer not a sum of distinct unit fractions with denominators ≤ N
Added: A101877

$f(N)$ in the problem is the inverse of A101877 ("a(n) = least k such that some set S ⊆ {1,…,k} has Σ 1/s = n"): $f(N)$ = least $n$ with $A101877(n) > N$. Terms $1, 6, 24, 65, …$ are the values of $N$ at which $f$ increments.

Verification: computed $f(N)$ for $N \le 24$ by exact subset-sum over $\mathrm{lcm}(1..N)$: $f = 2$ for $N<6$, $3$ for $6 \le N < 24$, $4$ at $N=24$ — i.e. increments at $N = 1, 6, 24$, matching $A101877(1..3)$. The next increment at $65$ is A101877(4) (Croot's paper cited on the problem page uses the same value).

Problem 829 -- $1_A * 1_A(n)$ for $A$ = cubes
Added: A025455, A025468

A025455 = number of partitions of $n$ into 2 positive cubes (unordered); A025468 = same with distinct cubes. The convolution in the problem is the ordered count, $= 2\cdot A025455(n) - [,n = 2m^3,]$; the growth question is unaffected by the normalisation.

Verification: recomputed both sequences for $n \le 6000$; they match the OEIS DATA lines term-for-term (offset 0), including the first values $\ge 2$ at $n = 1729, 4104$.

Problem 929 -- $S(k)$, least $x$ such that a positive-density set of $n$ has $n+1,…,n+k$ all divisible by primes $\le x$
Added: A058989, A048670, A049300

Whether $n+1,…,n+k$ are all covered by primes $\le p_m$ depends only on $n \bmod \prod_{p\le p_m} p$, so "positive density" is equivalent to "at least one $n$", and $S(k)$ is the inverse of A058989 ("largest number of consecutive integers each divisible by a prime ≤ the n-th prime"): $S(k)$ = least $p_m$ with $A058989(m) \ge k$. A048670 (Jacobsthal function at primorials) $= A058989 + 1$ and A049300 gives the starting points; both are cross-referenced from A058989.

Verification: brute-force longest covered run over one full period for the first 9 primes gives $1, 3, 5, 9, 13, 21, 25, 33, 39$, matching A058989(1..9).

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