Prime twin-center sigma-gcd values are two or three

For every natural number k, if k-1 and k+1 are prime and gcd(k, sigma(k)) is prime, then gcd(k, sigma(k))=2 or gcd(k, sigma(k))=3.

This is the second A394757 conjecture supplied in preregistration #15004. It determines the possible prime gcd values, complementing the existing divisibility-by-18 theorem.

The existing divisibility theorem constrains k but does not identify its prime sigma-gcd. The odd divisor-sum branch requires three to divide sigma(k).

An even sigma(k) gives two dividing the prime gcd, so that gcd equals two. For odd sigma(k), the existing divisibility-by-18 result excludes small centers, and the shared square exclusion gives k=2t² with t nonzero. Write t=2^b r with r odd. Coprime multiplicativity yields sigma(k)=sigma(2^(2b+1)) sigma(r²). The existing geometric-sum formula identifies the first factor as 2^(2b+2)-1. Since 4^(b+1) is congruent to one modulo three, this factor is divisible by three. The existing three_center theorem gives three dividing k. Thus three divides the prime gcd and that gcd equals three.

A natural twin-prime center whose sigma-gcd is prime and differs from both two and three would falsify the statement.

• Lean module: D5/S3/Arith/Robin/TwinPrimeSigmaGcdTwoOrThree.lean.
• Exact proposition: claim; universal proof: result : claim.
• Reused frozen declarations: TwinPrimeSigmaGcdDivisibility.even_center and KrizekTriangularSquareSigmaParity.sigma_odd_iff_square_or_twice_square.
• The proof is unbounded and does not assert that qualifying centers exist infinitely often.

• [proved] The universal statement in the Problem section.
• [open] Whether infinitely many centers satisfying these conditions exist. This involves unresolved twin-prime infinitude and the additional sigma-gcd restrictions; twin-prime infinitude alone does not settle those restrictions.

The OEIS attribution and the literature status are supplied by the preregistration. No network is available to independently reread the entry or extend its literature search. First-publication priority is not a kernel-checked conclusion. The Lean claim retains exactly the supplied quantifiers and hypotheses.