Open statements in core entries of the OEIS, II: further proofs and three refutations

The keyword "core" of the On-Line Encyclopedia of Integer Sequences (OEIS) marks 183 of its most fundamental sequences: the primes, the Fibonacci, Catalan, Motzkin and Schröder numbers, the partition numbers, the Euler and tangent numbers, the digits of pi, and so on. In a first report (Open statements in core entries of the OEIS: proofs, corrections and a refutation) we examined the statements labelled as conjectures, empirical or "apparent" in these entries. Here we widen the search to other wordings ("conjectures", "conjecturally", "appears to", "probably", ...) and re-examine every statement that was left open. We prove 33 further statements in 26 core entries, four of them after a small correction, and show that three conjectures are false. Among the proofs: three conjectures of P. Bala on the tangent numbers (eventual periodicity modulo k, Gauss congruences and their shifted form) and his continued fraction for them; two congruences of Bala for the Motzkin numbers; the sixth family of Bala's series for pi, which completes the first report; C. R. Greathouse's bound Delta(n) <= d(n)/2 for Hooley's function; Y. Yurramendi's statement on the values of Stern's sequence; statements of M. Granvik on the Möbius function and on lcm(1,...,n); and two numerical near-identities of S. Plouffe for the partition function, which we explain by exact formulas (they agree to 13 and 28 digits but are not identities). The three refutations are Bala's pure periodicity of the Euler numbers modulo k (false for k = 27; we prove the correct statement: the sequence is purely periodic modulo k exactly when no cube of an odd prime divides k) and two characterizations of the primes by G. Detlefs (smallest counterexamples 97921 = 181·541 and 219781 = 271·811). We also correct two remarks: a statement on West's stack-sorting map in the entry for the Schröder numbers, where the pattern 231 should read 312, and a remark on P. Barry's continued fraction for the Motzkin numbers. Most proofs are short and use classical tools (congruences for sums of powers, Lucas's and Kummer's theorems, Parseval's identity, the transformation of the Dedekind eta function, known continued fractions). Several statements that looked open turned out to be settled in other entries or in the literature; the report lists them so that the entries can be updated. Every result was checked by computer independently of the proofs. The verification program (verify_core2.py, Python with SymPy and mpmath, no external data, 61 checks) is included. The report was also checked by a separate AI review acting as referee, and all its corrections were applied. Contributions and use of AI: the author conceived and directed the work (the systematic examination of the core entries, its extension to other wordings and the re-examination of the statements left open). An AI assistant (Claude, by Anthropic), working under the author's direction, carried out the search, found and wrote the proofs and counterexamples, wrote the verification program and drafted the text. Details are given in the report. The report is provided in English and in Spanish.