Equivalently, k and k+1 have the same absolute value of abundance (or deficiency) with opposite signs.
Equivalently, s(k) + s(k+1) = k + (k+1), where s(k) is the sum of proper divisors of k (A001065).
If k is a 3-perfect number (A005820) and k+1 is a prime, then k is in the sequence. Of the 6 known 3-perfect numbers only 672 and 523776 have this property.
a(4) > 10^11, if it exists.
a(4) > 10^13, if it exists. - Giovanni Resta, May 30 2020