Bounded-gcd conjecture for shifted Kurepa factorial sequences

Counterexample for the conjectured shift $a=4$, with infinitely many failures.

Let

$$K_n={!n}=\sum_{j=0}^{n-1}j!, \qquad K_{n+1}-K_n=n!.$$

The source's other sequence is $F_n=K_{n+1}$, so its indices are shifted by one.

Direct exact evaluation gives

$$K_{17}=22\,324\,392\,524\,314\equiv13=-4\pmod{17},$$

and

$$K_{18}=378\,011\,820\,620\,314.$$

Therefore

$$\boxed{\gcd(K_{17}+4,K_{18}+4) =\gcd(K_{17}+4,17!)=34>2.}$$

Moreover, for every $n\ge17$,

$$K_n-K_{17}=\sum_{j=17}^{n-1}j!\equiv0\pmod{17}.$$

Hence $17\mid K_n+4$ for every $n\ge17$. Also $K_n$ is even for every $n\ge2$. Consequently

$$\boxed{34\mid\gcd(K_n+4,K_{n+1}+4) \qquad\text{for every }n\ge17.}$$

Equivalently, with the source's $F$-indexing,

$$34\mid\gcd(F_n+4,F_{n+1}+4) \qquad\text{for every }n\ge16.$$

More generally, there is a complete local criterion for an arbitrary integer shift:

$$\gcd(K_n+a,K_{n+1}+a)\le2 \quad\text{for every }n\ge2$$

if and only if

$$a\not\equiv2\pmod4 \quad\text{and}\quad a\not\equiv-K_p\pmod p \quad\text{for every odd prime }p.$$

Indeed,

$$\gcd(K_n+a,K_{n+1}+a)=\gcd(K_n+a,n!).$$

For any odd prime $p$ and $n\ge p$, factorial stabilization gives $K_n\equiv K_p\pmod p$; hence $p$ divides this gcd exactly when $a\equiv-K_p\pmod p$. Likewise $K_n\equiv2\pmod4$ for $n\ge4$, so divisibility of the gcd by $4$ occurs exactly when $a\equiv2\pmod4$. These exhaust all ways the gcd can exceed $2$.

The left-factorial values and the residue $K_{17}\bmod17=13$ are pre-existing data in OEIS A003422 and OEIS A100612, respectively. Applying that residue refutes both Conjecture 2 and the preceding shift-$4$ assertion in arXiv:2509.06077. The separate unshifted case $a=0$ remains the open Kurepa conjecture and is not claimed here.