[seqfan] characterizing A026416

[hv at crypt.org]

I've been meaning to investigate A026416 (and lead up to A026477) for 
a while, and finally made a start over the weekend.

I take it as read that membership of these sequences is determined
solely by prime signature, since that seems clear; I can elaborate 
on that if anyone disagrees.

Expressing prime signatures as multisets [a, b, c, ...], I find that
signatures in A026416 are easily characterized for simple cases:

  []
  [a] for a = 2 or a == 1 (mod 3)
  [a, b] for (a + b) == 1 (mod 3), except not [2, 2]

.. but then it starts to get more complex. For 3-element signatures, we
see these multisets (covering all sums up to 25):

apparently continuing patterns:
  [3a, 3b, 3c] for all except [3, 3, 3]

  [3a+1, 1, 1] from 3a+1 = 1

  [3a  , 2, 2] from 3a = 12
  [3a+2, 3, 2] from 3a+2 = 5
  [3a  , 5, 2] from 3a = 6
  [3a+2, 6, 2] from 3a+2 = 8
  [3a  , 8, 2] from 3a = 9
  [3a+2, 9, 2] from 3a+2 = 11
  [3a  , 11, 2] from 3a = 12

  [3a  , 5, 5] from 3a = 6
  (but not [3a+2, 6, 5]; eg [8, 6, 5] = [6, 3, 3] + [5, 2])

not obviously continuing patterns:
  [2, 2, 1]
  [3, 3, 1]
  [4, 4, 1]
  [4, 2, 2]
  [7, 2, 2]
  [5, 5, 3]
  [8, 5, 3]
  [4, 4, 4]
  [11, 8, 5] (plausibly the start of another pattern)
  [8, 8, 8] (plausibly the start of another pattern)

I'd be interested if someone is able to wrestle more regularity out of
that; I'm happy to generate results for larger sums or a larger number
of elements if anyone wants them.

The code I used is on github at [1]. It shouldn't be too hard to adapt
for A026477 (which was my original intent), but there seems little point
if this simpler case remains so hard to characterize.

If there are no new insights to be had, I'll try to summarize something
useful out of these findings in OEIS.

Hugo

[1] https://github.com/hvds/seq/blob/master/A026477/A026416