cp's OEIS Frontend

Conjecture 1: For all m > 1 there is always at least one Ulam number u(j) such that m < u(j) < 2m.

Conjecture 2: For all m > 4 there is always at least two Ulam numbers u(j), u(j+1) such that m < u(j) < u(j+1) < 2m.

This sequence illustrates how far these conjectures are oversatisfied.

Conjecture 1 implies that Ulam numbers form a complete sequence because u(1) = 1 and 2u(j) >= u(j+1).

Conjecture 2 implies that three consecutive Ulam numbers satisfies the triangle inequality because 2u(j) > u(j+2) > u(j+1) > u(j) and u(j) + u(j+1) > 2u(j) > u(j+2). It further implies that n consecutive Ulam numbers can always form an n-gon.

The percentage of odd Ulam numbers increases from 33.3333% (for N = 10) to 49.9962% (for N = 10^10) and approaches 50%.

The percentage of even Ulam numbers is decreasing from 66.66667% (for N = 10) to 50.00038% (for N = 10^10) and approaching 50%.

Numbers k such that k | A002858(k).

a(13) >= 10^8, if it exists.

Based on empirical data its seems that the Ulam numbers have a positive asymptotic density and that A002858(k) ~ 13.5... * k (see A307331 and A346216). If this is true, then this sequence is finite, and it is likely that there are no more terms.