bibkey: bala2024a344598 authors: Peter Bala year: 2024 title: OEIS A344598, conjectured alternating gcd-sum formula doi: null url: https://oeis.org/A344598 claim: “a(n) = Sum_{k=1..n} phi(k) * (floor(n/k)^2 - floor((n-1)/k)^2). Conjecture: a(n) = Sum_{k = 1..2n} (-1)^k * gcd(k, 4n). Cf. A344372. - Peter Bala, Jan 01 2024” strata_touched:

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OEIS A344598

Seiichi Manyama introduced the floor-square totient sequence on May 24, 2021. Peter Bala’s January 1, 2024 formula conjectures the alternating gcd sum for every positive index. This is a modest arithmetic identity.

The entry separately credits Daniel Weber with proofs of the divisor formulas a(n) = Sum_{d|n} phi(d)*(2*n/d - 1) and a(n) = Sum_{d|n} (2*d*tau(d) - sigma(d))*mu(n/d) in Mikhail Kurkov’s March 31, 2026 comment. Those attributions concern the divisor formulas; the alternating gcd formula is stated as a conjecture. The Lean definition uses natural-number division and truncated subtraction; the alternating sum takes values in the integers.

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• URL: https://oeis.org/A344598