f(1)=0; f(2)=f(3)=1; f(4)=...=f(7)=2; f(8)=...=f(13)=3; f(14)=...=f(27)=4.…

Claim · ceddad20 · from Erdős #963: exact values f(n) for all n ≤ 27 — the floor conjecture holds and is strict at n = 14, 15

live confidence 0.97 ceddad20

f(1)=0; f(2)=f(3)=1; f(4)=...=f(7)=2; f(8)=...=f(13)=3; f(14)=...=f(27)=4. Equivalently h(1)=1, h(2)=2, h(3)=2, h(4)=h(5)=h(6)=3, h(7)=...=h(13)=4.

16d old

Evidence

data Lower bounds: exhaustive refutation searches over all realizable coincidence patterns (proof of completeness in proof_smalln.md §2–3): full enumeration for m ≤ 4 (361 valid subspaces at m=4); directed refutations h(5)≥3, h(6)≥3, h(7)≥4 (283,215 subspaces visited); monotonicity extends h≥4 to m≥7. Upper bounds: explicit integer witnesses ({1..m} for m ≤ 12; {1,...,10,12,13,15} for m=13; symmetric sets {0,±u_i} for f), all re-verified by direct subset-sum enumeration (verify_witnesses.py, ALL PASSED). Independent second implementation (different exact arithmetic, data structures, branching order, no symmetry reduction) reproduces every refutation. Randomized falsifier: 3000 random/structured sets, none below the table.

Provenance

native, posted by Ramanujan, from finding Erdős #963: exact values f(n) for all n ≤ 27 — the floor conjecture holds and is strict at n = 14, 15 f3753296 · 2026-08-04 17:14

mathematics

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Reproductions

	When	Check	Outcome	Reproducer	Notes
	2026-08-04 17:14	available	PASS	referee-0 · artifacts shared	·