Open statements in core entries of the OEIS, III: Benford's law for the Bell numbers, Bala's congruences for the Motzkin numbers, and further proofs

The keyword "core" of the On-Line Encyclopedia of Integer Sequences (OEIS) marks 183 of its most fundamental sequences: the primes, the Fibonacci, Lucas, Catalan, Bell and Motzkin numbers, the Thue-Morse sequence, and so on. This is the third report on the statements labelled as conjectures, empirical, "apparent" or with similar wordings in these entries; the first two reports proved 61 of them and refuted four. Here we re-examine every statement that was left open, including those classified as famous or research-level problems. We prove 18 further statements in 12 core entries, four of them after a correction of their wording, we clarify two asymptotic statements, and we settle partially two more. The main results are: - N. J. A. Sloane's remark (2017) that the Bell numbers presumably satisfy Benford's law. We prove it from the saddle-point asymptotics of Moser and Wyman and van der Corput's second-derivative estimate; we have not found a proof in the literature. The same remark for the Catalan numbers follows from Weyl's criterion.- P. Bala's two remaining congruences for the Motzkin numbers, M(np^r-2) == -A005717(n-1) and M(np-2) == -A005773(n) (mod p). Both are special cases of a congruence valid for every prime p != 3 and every r >= 1, which also covers p == 2 (mod 3) with r >= 2 and p = 2, with a companion formula for p = 3. The proof uses the Frobenius map and the identity (1-x^2)/(1+x+x^2)^2 = d/dx [x/(1+x+x^2)].- The three observations recorded in the entry of the Lucas numbers on the primes that divide no Lucas number, two of them after a correction (one of them is Vinson's theorem on Pisano periods).- W. Schulte's general form of an identity of Dressler and van de Lune; R. J. Mathar's count of Lipschitz quaternions; four statements on the Thue-Morse sequence, including B. McEachen's frequency 2/3; C. Dement's statement relating A002316 and A002531; F. van Lamoen's statement relating Golomb's and Levine's sequences; the existence of G. McGarvey's limit for the Narayana-Zidek-Capell numbers; B. Cloitre's limit for the binary partitions, with the value 2 log 2 found by V. Kotesovec, as a corollary of de Bruijn's theorem; and M. Chu's statement on additive covers of {0, ..., n^2-1}.- Half of a conjecture of C. W. Wu on the Mersenne numbers (every 2^(m+1)-2^j-2^k-1 is composite when m = 2^n-1, n >= 3), and a reduction of a conjecture of T. Ordowski and G. Resta on the prime powers to a condition of Giuga type, which we prove when m is even or has at most three prime factors. Methodology: every statement is translated from words into a formula of a fixed type that the author's reasoning engine SyntheticMind can read, with a time budget depending on the type of computation; the engine checks it, splits it recursively into subgoals and records in a log of obstacles where and why a formula breaks. The methods suggested by that log (Frobenius congruences for coefficients of powers of polynomials, Thue-Morse identities and block frequencies, sets defined by power congruences, Benford's law by saddle-point asymptotics, ranks of apparition of Fibonacci numbers, among others) were added to the engine, which now proves 27 of the 121 statements of its test file, against 5 before this work. Every result was checked by computer independently of the proofs. The verification program (verify_core3.py, Python with SymPy and mpmath, 53 checks, about 20 seconds) is included. The report was also read by a separate AI system acting as referee, and all its corrections were applied. Contributions and use of AI: the author conceived and directed the work and its method (the translation of the statements into formulas for his engine, the time budgets, the log of obstacles, the recursive splitting into subgoals, the independent verification and the refereeing). An AI assistant (Claude, by Anthropic), working under the author's direction, found the proofs, implemented the corresponding methods in the engine, wrote the verification program and drafted the text. Details are given in the report. The report is provided in English and in Spanish.