
COMMENTS

Two circles are either disjoint or meet in two points. Tangential contacts are not allowed. A point belongs to exactly one or two circles. Three circles may not meet at a point. The circles may have different radii.
This is in the affine plane, rather than the projective plane.
Two arrangements are considered the same if one can be continuously changed to the other while keeping all circles circular (although the radii may be continuously changed), without changing the multiplicity of intersection points, and without a circle passing through an intersection point. Turning the whole configuration over is allowed.
Several variations are possible:
 straight lines instead of circles (see A241600).
 straight lines in general position (see A090338).
 curved lines in general position (see A090339).
 allow circles to meet tangentially but without multiple intersection points (begins 1, 5, ...); more terms are needed.
 again use circles, but allow multiple intersection points (also begins 1, 5, ...); more terms are needed.
 use ellipses rather than circles.
 a question from Walter D. Wallis: what if the circles must all have the same radius?
A representation for the diagrams in a250001.jpg (in the same order):
a(1) = 1: {{2}};
a(2) = 3: {{2, 3}, {2, 4}, {4, 6}};
a(3) = 14: {{2, 4, 8}, {2, 3, 6}, {2, 3, 4}, {2, 3, 5}, {4, 6, 5},
{4, 6, 15}, {2, 6, 9}, {4, 6, 12}, {2, 8, 12}, {30, 42, 70},
{?, ?, ?}, {?, ?, ?}, {15, 21, 35}, {?, ?, ?}}.
In lexicographic order:
a(3) = 14: {{2, 3, 4}, {2, 3, 5}, {2, 3, 6}, {2, 4, 8}, {2, 6, 9},
{2, 8, 12}, {4, 6, 5}, {4, 6, 12}, {4, 6, 15}, {15, 21, 35},
{30, 42, 70}, {?, ?, ?}, {?, ?, ?}, {?, ?, ?}}.
The smallest integers greater than 1 are used for the representation:
(p_1)^(a_1)*...*(p_m)^(a_m), where
0 <= a_i <= n, for 1 <= i <= m;
(a_1)+...+(a_m) > 0.
Could the Venn diagram interpretation (of the kwise, 1 <= k <= n, common divisors of k numbers from each subset) reveal a pattern?
Does this representation work for more complex diagrams? (End)
Comment from Jon Wild, Aug 25 2016. Once you get to n=5, geometric concerns mean that not all topologicallyconceivable arrangements are actually circledrawable. My program enumerated 16977 conceivable arrangements of 5 pseudocircles, and Christopher Jones and I together have figured out how to show that 26 of these arrangements are not actually circledrawable. So it seems that a(5) = 16951. This entry will be updated soon, and there will be a new sequence for the number of topologicallyconceivable arrangements. [The counts in this comment were amended by Jon Wild on Aug 30 2016. I apologize for taking so long to make the corrections here.  N. J. A. Sloane, Jun 11 2017]
a(n) <= 7*13^(binomial(n,3) + binomial(n,2) + 3n  1) is a (loose) upper bound, see Reddit link. I believe XkF21WNJ's reply shaves off a factor of 13^3 from this bound for all n > 1.  Linus Hamilton, Apr 14 2019
