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

1,2

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

Links

Table of n, a(n) for n=1..57.

Example

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).

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 *)

Prog

(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
(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

Crossrefs

The least such number is A062856(n).

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

Sequence in context: A332824 A053146 A056675 * A061799 A076868 A056793

Adjacent sequences: A062854 A062855 A062856 * A062858 A062859 A062860

Keyword

nonn

Author

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

Extensions

More terms from Don Reble, Nov 08 2001

Name clarified by Robert Israel, Jan 30 2017

Status

approved