A178776 a(n) is the largest number that appears twice in an n x n multiplication table of positive integers, disregarding the pairs that are obviously produced by the commutative property.

4, 4, 12, 12, 24, 36, 40, 40, 72, 72, 84, 120, 144, 144, 180, 180, 240, 252, 252, 252, 360, 400, 400, 432, 504, 504, 600, 600, 672, 672, 672, 840, 900, 900, 900, 936, 1120, 1120, 1260, 1260, 1320, 1440, 1440, 1440, 1680, 1764, 1800, 1800, 1872, 1872

Offset: 4

Author

Gary Yane, Jun 11 2010

Keywords

• nonn

Comments

It appears that all terms of this sequence are multiple of 4 (checked up to 10^7). - Michel Marcus, Aug 26 2013

Examples

			The first term is 4, which appears twice in a 4 x 4 multiplication table. a(4) = 2x2 = 4x1 = 4. For all n = prime a(n) = a(n-1) so a(5) = 4. a(6) = 12 = 2x6 = 3x4, a(7) = 12, a(8) = 24 = 3x8 = 4x6, a(9) = 36 = 4x9 = 6x6.
		

Programs

PARI

a(n) = {skeep = Set(); mmax = 0; for (i = 1, n, for (j = i, n, v = i*j; if (! setsearch(skeep, v), skeep = setunion(skeep, Set(v)), mmax = max(mmax, v)););); mmax;} \\ Michel Marcus, Aug 26 2013

PARI

findxy(n, d) = {mins = 2*n; if (#d % 2, nd = #d\2 +1, nd = #d/2); for (i = 1, nd, if ((s = d[i]+n/d[i]) < mins, mins = s; mini = i);); x = d[mini]; y = n/x; return (x*y*(x-1)*(y-1));}
lista(nn) = {lmmx = 4; print1(lmmx, ", "); for (n=5, nn, d = divisors(n); nmmx = findxy(n, d); lmmx = max(lmmx, nmmx); print1(lmmx, ", "););} \\ Michel Marcus, Aug 26 2013

Formula

Let xy = n be the factorization of n such that x+y is a minimum. We then have a(4)= 4 and a(n)= max{a(n-1), xy(x-1)(y-1)} for all n>4.

Extensions

Edited by N. J. A. Sloane, Jun 12 2010

Corrected by Michel Marcus, Aug 26 2013