A062857 Size of smallest square multiplication table which contains some number at least n times.

1, 2, 4, 6, 12, 12, 18, 20, 30, 30, 40, 40, 60, 60, 72, 72, 90, 90, 120, 120, 140, 140, 168, 168, 180, 180, 210, 210, 252, 252, 280, 280, 315, 315, 336, 336, 360, 360, 420, 420, 504, 504, 560, 560, 630, 630, 672, 672, 720, 720, 792, 792, 840, 840, 924, 924, 990

Offset: 1

Author

Ron Lalonde (ronronronlalonde(AT)hotmail.com), Jun 25 2001

Keywords

• nonn

Comments

a(n) is the least number m such that there exists k with 1 <= k <= m^2 such that k has at least n divisors t with k/m <= t <= m. - Robert Israel, Jan 30 2017

Examples

			a(7)=18 because the 18 X 18 multiplication table is the smallest to contain a product of frequency 7 (namely the number A062856(7)=36).
		

Crossrefs

The least such number is A062856(n).

Cf. A027424, A062851, A062854, A062855, A062856, A062858, A062859.

Programs

MATLAB

N = 1000; % to get all terms with a(n) <= N
M = sparse(1,N^2);
A(1) = 1;
imax = 1;
for k = 2:N
  M(k*[1:k-1]) = M(k*[1:k-1])+2;
  M(k^2) = 1;
  newimax = max(M);
  A(imax+1:newimax) = k;
  imax = newimax;
end
A  % Robert Israel, Jan 30 2017

Mathematica

a[1] = 1; a[n_] := a[n] = For[m = a[n-1], True, m++, T = Table[i j, {i, m}, {j, m}] // Flatten // Tally; sel = SelectFirst[T, #[[2]] >= n&]; If[sel != {}, Print[n, " ", m, " ", sel[[1]]]; Return[m]]];
Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Mar 25 2019 *)

Python

from itertools import count
from collections import Counter
def A062857(n):
    if n == 1: return 1
    c = Counter()
    for m in count(1):
        for i in range(1,m):
            ij = i*m
            c[ij] += 2
            if c[ij]>=n:
                return m
        c[m*m] = 1 # Chai Wah Wu, Oct 16 2023

Extensions

More terms from Don Reble, Nov 08 2001

Name clarified by Robert Israel, Jan 30 2017