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

4,1

Comments

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

Marcus's comment is true. Indeed, the proof follows by induction and by noticing that both x*(x-1) and y*(y-1) are even. - Sela Fried, Nov 15 2025

Links

Table of n, a(n) for n=4..53.

Formula

Let x*y = 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), x*y*(x-1)*(y-1)} for all n>4.

Example

The first term is 4, which appears twice in a 4 X 4 multiplication table. a(4) = 2*2 = 4*1 = 4. For all n = prime a(n) = a(n-1) so a(5) = 4. a(6) = 12 = 2*6 = 3*4, a(7) = 12, a(8) = 24 = 3*8 = 4*6, a(9) = 36 = 4*9 = 6*6.

Prog

(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

Crossrefs

Sequence in context: A273491 A168398 A281913 * A282503 A323189 A121189

Adjacent sequences: A178773 A178774 A178775 * A178777 A178778 A178779

Keyword

nonn

Author

Gary Yane, Jun 11 2010

Extensions

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

Corrected by Michel Marcus, Aug 26 2013

Status

approved