comment on A093782

[Richard Mathar]

Dear Seq-aficionados, SeqFans, or other technologies of air-conditioners,

I think A093782 ought be changed from
5, 0, 7, 19, 127, 61, 2, 31, 97, 13, 23, 269, 53, 239, 563, 569
to
5, 0, 17, 19, 127, 61, 2, 31, 97, 13, 23, 269, 53, 239, 181, 449, 541, 11

The reason to change a(3) from 7 to 17 is trivial: the EM sequence that
starts with 7 is
   7,2 (factorization of 7+1=8),3 (factorization of 7*2+1=15),43
  (factorization of 7*2*3+1=43),13,53,5
so the 5 appears not at the third place but the seventh. We do not
set a(7)=7 because a(7)=2 is a better (smaller=earlier) solution.
The sequence for 17 is
17,2,5 (factorization of 17*2+1=35)
where the 5 is at the third place, therefore a(3)=17. This can be booked
as a typo.

The reason to change a(15) from 563 to 181 and a(16) from 569 to 449,
is that, although both have EM sequences with 5 at the correct
position, the sequences that start at 181 and 449 have smaller starting
values. So if one rephrases the definition more precisely as "a[n] is the
*smallest* initial value (=prime) for...", the 181 and 449 are better solutions,
with EM sequences
181,2,3,1087,73,7,29,151,61,98689,11,10929259909,678859,97,5
and
449,2,29,3,7,349,190861819,166273,16091,11,3807491,53,17,313,23,5

a(16)=449 is still conjectural since I computed the sequence for 383
only to its 15th but not to its 16th term, so 383 remains a better
(=smaller than 449) candidate for a(16).

The EM sequence for 11 is
11,2,23,3,7,13,10805892983887, 73, 6397, 19, 489407, 2753, 87491, 18618443, 5
with the 5 at the 18th place, so a(18)=11.

RJM