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Assuming that every even number above 6 is the sum of 2 distinct prime numbers, p + q (a slightly stronger version of the Goldbach conjecture), then every odd number m above 7 is of the form 1 + p + q, so A001065(p*q) = m. If this is true, then 5 is the only odd untouchable number (A005114).

Alanen (1972) suggested the study of odd numbers that are being "touched" only by Goldbach solutions, i.e., odd numbers k such that there is no solution m to A001065(m) = k which is not a squarefree semiprime. He suggested that perhaps these numbers deserved to be called "almost untouchable" numbers.

Abundant numbers that are not the sum of any subset of their aliquot divisors, and are also not the sum of all the aliquot divisors of any other number.

Let s^m(k) denote the m-th iterate of s(k) = sigma(k) - k. n-untouchable numbers are the numbers that lie in the image of s^(n-1)(k), but not in the image of s^n(k).

a(19) = 2255 is the least term that is not divisible by 3.

k is in sequence iff k can never be reached when iterating the map x -> A001065(x) starting with any odd number m.

Assuming the stronger version of Goldbach conjecture, iff k is in the sequence, there are infinitely many odd numbers whose aliquot sequence contain k.

Supersequence of A005114 (except 5), A283152, A284147, A284156, A284187, ..., and untouchable perfect numbers (28, 137438691328, ...), untouchable amicable numbers (A238382), untouchable sociable numbers.

Highly touchable numbers k have a record number of solutions x to A001065(x) = k, while untouchable numbers k have no solution to this equation.