HELP WANTED: Computing F(n) for Erdős problem #425 (multiplicative Sidon)

[arcoretx]

Problem #425 defines:

$F(n)$ is the maximum possible size of a subset $A \subseteq {1,\ldots,n}$ such that the products $ab$ are distinct for all $a < b$ in $A$.

The natural integer sequence $F(1), F(2), F(3), \ldots$ does not currently appear in OEIS under any obvious search term (multiplicative Sidon, $B_2^*$, distinct products, etc.). The published multiplicative-Sidon literature (Erdős 1938; Erdős–Sárközy–Szemerédi; Pach–Sándor; Liu–Pach) focuses on asymptotic bounds, not tabulated small-$n$ values. This sequence seems worth submitting to OEIS as a new entry.

Computed values

I computed $F(n)$ for $n = 1 \ldots 80$ by encoding the problem as an integer program:

• Variables: $x_i \in {0,1}$ for $i = 1, \ldots, n$.
• Objective: maximize $\sum_i x_i$.
• Constraints: for each $(a,b,c,d)$ with $a < b$, $c < d$, $(a,b) \neq (c,d)$, $ab = cd$, all in ${1,\ldots,n}$: $x_a + x_b + x_c + x_d \leq 3$.

The objective values were obtained from three independent implementations:

1. CBC MILP (via PuLP).
2. Google OR-Tools CP-SAT.
3. Pure-Python branch-and-bound (different solving paradigm; only tractable up to $n=30$).

Cross-verification status:

• $n = 1..30$: all three implementations agree.
• $n = 31..80$: CBC MILP and CP-SAT agree.

| n | F(n) | π(n) | F-π |
|---|---|---|---|
| 1 | 1 | 0 | 1 |
| 2 | 2 | 1 | 1 |
| 3 | 3 | 2 | 1 |
| 4 | 4 | 2 | 2 |
| 5 | 5 | 3 | 2 |
| 6 | 5 | 3 | 2 |
| 7 | 6 | 4 | 2 |
| 8 | 6 | 4 | 2 |
| 9 | 7 | 4 | 3 |
| 10 | 7 | 4 | 3 |
| 11 | 8 | 5 | 3 |
| 12 | 9 | 5 | 4 |
| 13 | 10 | 6 | 4 |
| 14 | 10 | 6 | 4 |
| 15 | 10 | 6 | 4 |
| 16 | 11 | 6 | 5 |
| 17 | 12 | 7 | 5 |
| 18 | 12 | 7 | 5 |
| 19 | 13 | 8 | 5 |
| 20 | 13 | 8 | 5 |
| 21 | 13 | 8 | 5 |
| 22 | 14 | 8 | 6 |
| 23 | 15 | 9 | 6 |
| 24 | 15 | 9 | 6 |
| 25 | 16 | 9 | 7 |
| 26 | 16 | 9 | 7 |
| 27 | 16 | 9 | 7 |
| 28 | 16 | 9 | 7 |
| 29 | 17 | 10 | 7 |
| 30 | 17 | 10 | 7 |
| 31 | 18 | 11 | 7 |
| 32 | 19 | 11 | 8 |
| 33 | 19 | 11 | 8 |
| 34 | 19 | 11 | 8 |
| 35 | 20 | 11 | 9 |
| 36 | 20 | 11 | 9 |
| 37 | 21 | 12 | 9 |
| 38 | 21 | 12 | 9 |
| 39 | 21 | 12 | 9 |
| 40 | 21 | 12 | 9 |
| 41 | 22 | 13 | 9 |
| 42 | 23 | 13 | 10 |
| 43 | 24 | 14 | 10 |
| 44 | 24 | 14 | 10 |
| 45 | 24 | 14 | 10 |
| 46 | 24 | 14 | 10 |
| 47 | 25 | 15 | 10 |
| 48 | 25 | 15 | 10 |
| 49 | 26 | 15 | 11 |
| 50 | 26 | 15 | 11 |
| 51 | 26 | 15 | 11 |
| 52 | 27 | 15 | 12 |
| 53 | 28 | 16 | 12 |
| 54 | 28 | 16 | 12 |
| 55 | 28 | 16 | 12 |
| 56 | 29 | 16 | 13 |
| 57 | 29 | 16 | 13 |
| 58 | 29 | 16 | 13 |
| 59 | 30 | 17 | 13 |
| 60 | 30 | 17 | 13 |
| 61 | 31 | 18 | 13 |
| 62 | 31 | 18 | 13 |
| 63 | 31 | 18 | 13 |
| 64 | 31 | 18 | 13 |
| 65 | 32 | 18 | 14 |
| 66 | 32 | 18 | 14 |
| 67 | 33 | 19 | 14 |
| 68 | 33 | 19 | 14 |
| 69 | 33 | 19 | 14 |
| 70 | 33 | 19 | 14 |
| 71 | 34 | 20 | 14 |
| 72 | 35 | 20 | 15 |
| 73 | 36 | 21 | 15 |
| 74 | 36 | 21 | 15 |
| 75 | 36 | 21 | 15 |
| 76 | 36 | 21 | 15 |
| 77 | 36 | 21 | 15 |
| 78 | 37 | 21 | 16 |
| 79 | 38 | 22 | 16 |
| 80 | 38 | 22 | 16 |

Sequence (comma-separated):

1,2,3,4,5,5,6,6,7,7,8,9,10,10,10,11,12,12,13,13,13,14,15,15,16,16,16,16,17,17,18,19,19,19,20,20,21,21,21,21,22,23,24,24,24,24,25,25,26,26,26,27,28,28,28,29,29,29,30,30,31,31,31,31,32,32,33,33,33,33,34,35,36,36,36,36,36,37,38,38

Cross-check: asymptotic behavior

Problem #425 asks about the constant $c$ in $F(n) = \pi(n) + (c + o(1)) \cdot n^{3/4} / (\log n)^{3/2}$. The $F(n) - \pi(n)$ column above grows smoothly and is consistent with that asymptotic shape (the small-$n$ regime is far from asymptotic, so this is a sanity check, not an estimate of $c$).

Caveats

• The computation was AI-assisted (Claude). Per the project's contributing guidelines, AI-generated submissions to OEIS are forbidden, so this is not a submission. The values would need independent human re-derivation before any OEIS submission.
• I did not exhaustively rule out a variant of $F(n)$ already in OEIS under different definitions (e.g., the variant allowing $a \leq b$, which counts $a \cdot a$ products and gives a slightly different sequence). My searches did not surface one, but a more thorough check before any OEIS submission would be valuable.
• Code (Python; CBC and CP-SAT solvers) is reproducible and available on request.

(Filed alongside PRs #283-289 and #292-296, and issues #290-291 and #297-298 in the same Tier-1 OEIS-linking pass.)

[jmullings]

Here is my proof candidate for you!