[seqfan] Re: Numbers > 1 not multiple nor a sum of any other terms.

[Neil Sloane]

David Seal makes some excellent comments and indeed the definition should
be clarified in the way he suggests.

As for "lexicographically earliest sequences " versus "a(n) is the smallest
number such that ...", remember these are different conditions: the former
is a global condition, the latter is a local condition. It probably doesn't
matter here, so let's use "greedy", which is a lot simpler.

The point is that "lex earliest" allows backtracking, whereas "a(n) is
smallest" uses the greedy algorithm and doesn't let you correct mistakes if
you run into a dead end. Example: A327762 uses greedy alg and dies after 56
terms, while A327460 is lex earliest and is infinite.

Jonathan, will you make the necessary changes to your submission?

Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com



On Sun, Dec 1, 2019 at 10:23 AM Christian Lawson-Perfect <
christianperfect at gmail.com> wrote:

> A similarly naive Python script produced the following 31 terms in a few
> seconds, then ran out of memory:
>
> 2, 3, 7, 11, 17, 25, 59, 67, 185, 193, 563, 571, 1697, 1747, 5141, 5149,
> 11995, 25727, 27439, 78893, 82345, 240131, 243583, 723845, 727297, 2174987,
> 2178439, 6530119, 6530123, 13061947, 19590377
>
> On Sun, 1 Dec 2019 at 09:46, jnthn stdhr <jstdhr at gmail.com> wrote:
>
> > Hi all.
> >
> > I didn't find this in the db, and superseeker had no suggestions.
> > https://oeis.org/A330070 is the sequence of numbers that are neither a
> sum
> > nor a multiple of smaller terms, and starts:
> >
> > 2, 3, 7, 11, 17, 25, 59, 67, 185, 193, 563, 571, 1697, 1747, 5141, 5149,
> > 11995, 25727, 27439, 78893, 82345, 240131, 243583,...
> >
> > Example:  a(6) = 25, because 25 = 5 x 5, and 5 is not in the sequence,
> and
> > no combination of 2, 3, 7, 11, and 17 sum to 25.
> >
> > The divisors (d > 1) of composite terms are:
> >
> > 25 [5]
> > 185 [5, 37]
> > 5141 [53, 97]
> > 5149 [19, 271]
> > 11995 [5, 2399]
> > 25727 [13, 1979]
> > 27439 [23, 1193]
> > 82345 [5, 16469, 43, 1915, 215, 383]
> >
> > Based on my original idea below (composites with this property), my
> > conjecture is that composite terms > 25  will only have either two or six
> > non-trivial divisors.
> >
> > My code takes ten+ minutes to find the first 21 terms.  The "is n a
> > multiple" test is efficient enough, just test if any divisors of n are in
> > the sequence.  As for sums, naively, I am storing all combinations, which
> > means in the worst case ~2^n sums must be checked.  Any ideas on how to
> > improve on this?
> >
> > A330071 will be the composite-only version, if that seems appropriate.
> > 4, 6, 9, 14, 21, 22, 38, 106, 111, 118,123, 465, 470,1394, 1405, 4193,
> > 4209, 9446,13289, 22258, 26101, 70617, 79959, ...
> >
> > divisors > 1:
> > 4 [2]
> > 6 [2, 3]
> > 9 [3]
> > 14 [2, 7]
> > 21 [3, 7]
> > 22 [2, 11]
> > 38 [2, 19]
> > 106 [2, 53]
> > 111 [3, 37]
> > 118 [2, 59]
> > 123 [3, 41]
> > 465 [3, 155, 5, 93, 15, 31]
> > 470 [2, 235, 5, 94, 10, 47]
> > 1394 [2, 697, 17, 82, 34, 41]
> > 1405 [5, 281]
> > 4193 [7, 599]
> > 4209 [3, 1403, 23, 183, 61, 69]
> > 9446 [2, 4723]
> > 13289 [97, 137]
> > 22258 [2, 11129, 31, 718, 62, 359]
> > 26101 [43, 607]
> > 70617 [3, 23539]
> > 79959 [3, 26653, 11, 7269, 33, 2423]
> >
> >  For this one, super seeker suggested  (lgdegf)
> > 81-216*a(n)+216*a(n)^2-96*a(n)^3+16*a(n)^4.
> >
> > Jonathan
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>