New sequences related to A000028/A000379

[David W. Wilson]

In recent discussion, I learned that {A, B} = {A000028, A000379} (as they
now stand) are the unique pair of sets complementary with respect to the
positive integers such that

	p(n) = |{x : x, y in A, x < y, xy = n}| = |{x : x, y in B, x < y, xy
= n}|

for all n >= 0.

Similarly, {A, B} = {A000069, A001969} are the unique pair of sets
complementary with respect to the nonnegative integers such that

	q(n) = |{x : x, y in A, x < y, x + y = n}| = |{x : x, y in B, x < y,
x + y = n}|

for all n >= 0.

We find that

p = (
    0,0,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,1,
    1,0,1,1,0,0,0,0,1,1,1,0,2,0,1,1,1,0,0,1,0,1,1,0,1,0,1,1,1,1,0,0,1,1,0,
    0,1,0,1,1,1,1,0,0,2,0,1,0,1,1,1,1,0,0,1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0
    ...
)

q = (
    0,0,0,1,0,1,1,0,1,2,1,1,2,1,1,4,1,2,3,1,3,3,2,4,3,2,3,5,2,5,5,0,5,6,3,5,
 
5,3,4,8,4,4,6,5,5,7,6,4,7,6,5,9,5,7,8,4,7,10,7,5,10,5,5,16,5,6,11,5,9,11,
    8,8,10,8,8,13,7,11,12,4,12,12,8,13,10,9,11,12,10,12,12,9,13,11,10,16,11
    ...
)

which may be OEISworthy.