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

[jnthn stdhr]

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