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

[Christian Lawson-Perfect]

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
>
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>