A239276 Smallest start for n consecutive numbers such that all the products of any two distinct numbers are distinct.

1, 1, 1, 1, 1, 4, 5, 7, 9, 13, 13, 22, 22, 25, 25, 37, 37, 51, 51, 57, 67, 73, 73, 92, 92, 100, 113, 121, 121, 145, 145, 172, 183, 211, 211, 211, 243, 256, 281, 289, 289, 326, 331, 346, 369, 385, 385, 426, 426, 443, 469, 487, 487, 533, 533, 581, 581, 601, 601

Offset

1,6

Comments

a(n-1) <= a(n) <= n^2.

Links

Steve Butler, Table of n, a(n) for n = 1..500

Example

For n=6 we have a(n)=4; 1 is impossible because 1*6=2*3, 2 is impossible because 2*6=3*4, and 3 is impossible because 3*8=4*6; however, the products of pairs of distinct numbers from {4,5,6,7,8,9}, i.e., 20,24,28,30,32,35,36,40,42,45,48,54,56,63,72, are all distinct. (Note that we do not count 6*6=4*9 since 6*6 does not involve distinct terms.)

Mathematica

a[1]=1; a[n_] := a[n] = Block[{k = a[n-1]}, While[Min@ Differences@ Sort[Times @@@ Subsets[Range[k, n+k-1], {2}]] == 0, k++]; k]; Array[a, 60] (* Giovanni Resta, Mar 14 2014 *)

Prog

(Sage)
def find_start(n):
    q=1
    while True:
        L={}
        advance=True
        for i in range(n-1):
            for j in range(i+1, n):
                if (q+i)*(q+j) not in L:
                    L[(q+i)*(q+j)]=1
                else:
                    advance=False
                    break
            if not advance:
                break
        else:
            return q
        q+=1

Crossrefs

Cf. A239277.

Sequence in context: A297838 A032360 A117150 * A284132 A263427 A052147

Adjacent sequences: A239273 A239274 A239275 * A239277 A239278 A239279

Keyword

nonn

Author

Steve Butler, Mar 13 2014

Status

approved