A134320 Positive integers with more non-isolated divisors than isolated divisors.

2, 4, 6, 12, 20, 30, 42, 90

Offset: 1

Author

Leroy Quet, Oct 20 2007

Keywords

• nonn

Comments

A divisor k of n is isolated if neither k-1 nor k+1 divides n (see A133779, A132881).

Is this sequence finite? One can show that, with the exception of a(2) = 4, all terms of this sequence must be of the form m*(m+1), oblong numbers, A002378.

Comments from Hugo van der Sanden, Oct 30 2007 and Oct 31 2007: (Start) A quick program to check found no other example up to 3e6, which certainly suggests it is not just finite but complete.

Partial proof: if adjacent integers k, k+1 both divide n then since they are coprime we also have that k(k+1) divides n, so k < sqrt(n).

I.e. the largest non-isolated factor a number can have is ceiling(sqrt(n)).

Since the divisors are symmetrically disposed around the square root, we have: if n is nonsquare, to be in this sequence it must be an oblong number, with all divisors below the square root non-isolated; if n is square, say n = m^2, then we have n divisible by m^2(m-1), so we require m-1 = 1.

So the only square entry is n = 4.

It remains to prove that there is no oblong number greater than 9*10 that avoids isolated divisors below the square root. (End)

Examples

			The divisors of 42 are 1,2,3,6,7,14,21,42. Of these, 1,2,3,6,7 are non-isolated divisors and 14,21,42 are isolated divisors. There are more non-isolated divisors (5 in number) than isolated divisors (3 in number), so 42 is in the sequence.
		

Crossrefs

Cf. A134321, A134322.