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Mathematical analysis demonstrates a sharp upper bound of 9n/4 for shifted divisor gcd sums across integers, highlighting 36 as the unique equality point.
For every integer n >= 2, defineA(n) = sumd|n gcd(d-1, n/d-1).We prove the sharp inequality A(n) <= 9n/4. Equality holds if and onlyif n = 36. The argument is elementary apart from an exact finiteverification for 2 <= n <= 1,340,963; complete source code for thatverification is supplied. We also record a reformulation as a restrictedshifted gcd sum modulo n-1.
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