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

[jnthn stdhr]

@Fred Lunnon:  The generating functions superseeker returns always baffle
me.   Would you mind adding to the Formula section of A330071?

@David Seal:  Thanks for the feedback on the name, and the wiki links.
Current name is now "a(n) is the smallest integer > 1 such that a(n) cannot
be expressed as the sum of any combination of distinct smaller terms, and
is not a multiple of any smaller term.

@Christian Lawson-Perfect:  Your Python code is clearly more efficient than
my Python code.  Would you like to add to the Extensions and Program
sections? Of the eight additional terms you found, four are composite and
have either two or six non-trivial divisors, so my conjecture holds, so far.

On Sun, Dec 1, 2019 at 7:23 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/
> >
>
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