A347498 Least k such that there exists an n-element subset S of {1,2,...,k} with the property that all products i * j are distinct for i <= j.

1, 2, 3, 5, 6, 7, 9, 11, 13, 15, 17, 19, 20, 23, 25, 28, 29, 31, 33, 37, 40, 41, 42, 43, 47, 51, 53, 55, 57, 59, 61, 67, 69, 71, 73, 75, 79, 83

Offset: 1

Author

Peter Kagey, Sep 03 2021

Keywords

• nonn
• more

Comments

a(n) <= A066720(n) and a(n+1) >= a(n) + 1

Examples

			   n | example set
-----+-------------------------------------------------------
   1 | {1}
   2 | {1, 2}
   3 | {1, 2, 3}
   4 | {1, 2, 3, 5}
   5 | {1, 3, 4, 5,  6}
   6 | {1, 3, 4, 5,  6,  7}
   7 | {1, 2, 5, 6,  7,  8,  9}
   8 | {1, 2, 5, 6,  7,  8,  9, 11}
   9 | {1, 2, 5, 6,  7,  8,  9, 11, 13}
  10 | {1, 2, 5, 7,  8,  9, 11, 12, 13, 15}
  11 | {1, 2, 5, 7,  8,  9, 11, 12, 13, 15, 17}
  12 | {1, 2, 5, 7,  8,  9, 11, 12, 13, 15, 17, 19}
  13 | {1, 5, 6, 7,  9, 11, 13, 14, 15, 16, 17, 19, 20}
  14 | {1, 2, 5, 7, 11, 12, 13, 16, 17, 18, 19, 20, 21, 23}
For n = 4, the set {1,2,3,4} does not have distinct products because 2*2 = 1*4. However, the set {1,2,3,5} does have distinct products because 1*1, 1*2, 1*3, 1*5, 2*2, 2*3, 2*5, 3*3, 3*5, and 5*5 are all distinct.
		

Crossrefs

Analogous for sums: A003022 and A227590.

Cf. A066720, A338006, A347499.

Programs

Mathematica

Table[k=1;While[!Or@@(Length[s=Union[Sort/@Tuples[#,{2}]]]==Length@Union[Times@@@s]&/@Subsets[Range@k,{n}]),k++];k,{n,12}] (* Giorgos Kalogeropoulos, Sep 08 2021 *)

Python

from itertools import combinations, combinations_with_replacement
def a(n):
    k = n
    while True:
        for Srest in combinations(range(1, k), n-1):
            S = Srest + (k, )
            allprods = set()
            for i, j in combinations_with_replacement(S, 2):
                if i*j in allprods: break
                else: allprods.add(i*j)
            else: return k
        k += 1
print([a(n) for n in range(1, 15)]) # Michael S. Branicky, Sep 08 2021

Formula

a(n) = min {k >= 1; A338006(k) = n}. - Pontus von Brömssen, Sep 09 2021

Extensions

a(15)-a(20) from Michael S. Branicky, Sep 08 2021

a(21)-a(38) (based on the terms in A338006) from Pontus von Brömssen, Sep 09 2021