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

[jnthn stdhr]

I just tried enclosing my code from A330070 in a for-loop that increments
the initial starting value (2,3,4,5,...) and get the following:  (n,  [seq
values])

(2, [2, 3, 7, 11, 17, 25, 59, 67, 185, 193, 563, 571, 1697, 1747])
(3, [3, 4, 5, 11, 13, 34, 71, 73, 179, 223, 397, 466, 1333, 1334, 2837,
4162])
(4, [4, 5, 6, 7, 19, 27, 47, 67, 179, 181, 374, 554, 922, 2218, 2398])
(5, [5, 6, 7, 8, 9, 31, 33, 34, 101, 209, 241, 652, 1311, 1343, 3508, 4642])
(6, [6, 7, 8, 9, 10, 11, 39, 85, 86, 87, 89, 349, 782, 785, 787, 2356,
2357])
(7, [7, 8, 9, 10, 11, 12, 13, 67, 68, 69, 71, 73, 346, 412, 1171, 1172,
1174, 1175, 4691, 4695])
(8, [8, 9, 10, 11, 12, 13, 14, 15, 76, 161, 163, 164, 166, 167, 827, 1811,
1813, 1814, 1817])
(9, [9, 10, 11, 12, 13, 14, 15, 16, 17, 109, 111, 113, 114, 115, 116, 683,
685, 2051, 2053])
(10, [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 125, 261, 262, 263, 265, 267,
268, 269, 1861, 2127])
(11, [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 166, 167, 172, 173, 174,
175, 1033, 1201, 1202, 3605, 3607])
(12, [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 186, 385, 386, 387,
388, 389, 393, 394, 395, 3121, 3123])
(13, [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 235, 236, 237,
239, 241, 243, 244, 245, 246, 2173, 2411, 4582])
(14, [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 259, 533,
534, 535, 536, 537, 538, 539, 541, 542, 543, 545])
(15, [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 316, 317,
318, 321, 326, 327, 328, 329, 2587, 2588, 2589, 2590, 2593])
(16, [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 344,
705, 706, 707, 708, 709, 710, 711, 712, 715, 716, 717, 718, 719])
(17, [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33,
409, 410, 411, 412, 413, 415, 417, 419, 421, 422, 423, 424])
(18, [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34,
35, 461, 463, 466, 467, 469, 470, 471, 472, 473, 474, 478])
(19, [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35,
36, 37, 514, 515, 516, 517, 519, 521, 523, 524, 526, 530, 531])
(20, [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36,
37, 38, 39, 571, 573, 574, 577, 579, 581, 582, 583, 584, 586, 587])
.
.
.
Here, we can see the first n terms are consecutive, followed by a first
gap.  The sequence of numbers that follow the first gap is:

7 11 19 31 39 67 76 109 125 166 186 235 259 316 344,...

Should I add these?  If so, how?

On Sun, Dec 1, 2019 at 6:23 PM jnthn stdhr <jstdhr at gmail.com> wrote:

> @Neil Sloane: The name in A330071 isn't correct, as the terms in that seq.
> aren't found in A330070, no?
>
> On Sun, Dec 1, 2019 at 4:51 PM Neil Sloane <njasloane at gmail.com> wrote:
>
>> I've edited both A330070 and A330071.
>>
>> 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 1:12 PM Neil Sloane <njasloane at gmail.com> wrote:
>>
>> > 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/
>> >>
>> >
>>
>> --
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>