[seqfan] Slowest possible pace at which iterated n + sopfr(n) can reach a given milestone?

Alonso Del Arte alonso.delarte at gmail.com
Mon Jul 18 18:58:50 CEST 2011

In pondering sequences like A096461 and A192896, it is clear that the
fastest possible pace to a given milestone is if it hits a prime every other
term on the way to the milestone. If a(1) = p, then a(2) = 2p
(composite) and a(3) = 3p + 2, which could be another prime and thus the
cycle continues. If a(n) is composite, then clearly a(n + 2) < 3a(n) + 2.

But what is the slowest possible pace? If a(1) = 2^k, then a(2) = 2^k + ,k,
but that won't be another power of two if k > 2. For example, with 2^32, the
next term is 4294967328, a very small increase over the previous term, but
then it contains a somewhat larger prime among its factors, 87211, and a
larger one still on the next iteration. This takes almost thirty steps to
reach the milestone of 2^33, which is very slow compared to 2^32 + 2 (just 6
steps) but I wonder if a slower pace is possible.


P.S. As a practical application to the OEIS, the milestone would probably be
about 200 for the smallest a(1)s.

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