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The complexity of a natural number n is the minimum number of ones required to write an arithmetic expression for n using ones, addition, multiplication and parentheses. It is denoted by $\left\| n \right\|$. The integer complexity has a corresponding sequence in The On-Line Encyclopedia of Integer Sequences: A005245. Jānis Iraids has calculated the sequence up to $n=10^{12}$ and the data is available in an interactive form.

Equivalently,

$\left\| n \right\| = \min_{a+b = n \vee a\cdot b = n} {(\left\| a \right\| + \left\| b \right\|)}$.

Logarithmic complexity

Since the integer complexity of n is bounded by $c\cdot\log_3{n}$, the logarithmic complexity $\left\| n \right\|_{log} = \frac{\left\| n \right\|}{\log_3 {n}}$ is of interest as a sort of efficiency measure of n.

Distribution of logarithmic complexity

Logarithmic complexity distribution for numbers of equal complexity.

The "best" numbers

The numbers on the left side of the logarithmic complexity distribution curves are simple and their expressions are known. These numbers are also known as sequence A000792 on The On-Line Encyclopedia of Integer Sequences.

The "worst" numbers

The numbers represented on the right side of the logarithmic complexity distribution curves are more complex. The number represented by the rightmost nonzero column is the smallest number of the given complexity. These numbers are also known as sequence A005520 on The On-Line Encyclopedia of Integer Sequences.

It has been conjectured that starting from the 26th all these numbers are primes.

Some numerical data

This table summarizes what is known about the "best" and "worst" numbers as well as the distribution of numbers of equal complexity.

$n$	A005520(n)	$\left\|A005520(n)\right\|_{\log}$	PrimeQ[A005520(n)]	$\underset{m}{\operatorname{max}} \, \{m | A005520(n) \equiv -1 \pmod{m!}\}$	A000792(n)	A133374(n)	A005421(n)
1	1	-	False	2	1	0	1
2	2	3.17	True	0	2	0	1
3	3	3.00	True	2	3	0	1
4	4	3.17	False	0	4	0	1
5	5	3.41	True	3	6	1	2
6	7	3.39	True	2	9	2	3
7	10	3.34	False	0	12	2	2
8	11	3.67	True	3	18	7	6
9	17	3.49	True	3	27	10	6
10	22	3.55	False	0	36	14	7
11	23	3.85	True	4	54	31	14
12	41	3.55	True	3	81	40	16
13	47	3.71	True	4	108	61	20
14	59	3.77	True	3	162	103	34
15	89	3.67	True	3	243	154	42
16	107	3.76	True	3	324	217	56
17	167	3.65	True	4	486	319	84
18	179	3.81	True	3	729	550	108
19	263	3.75	True	4	972	709	152
20	347	3.76	True	3	1458	1111	214
21	467	3.75	True	3	2187	1720	295
22	683	3.70	True	3	2916	2233	398
23	719	3.84	True	6	4374	3655	569
24	1223	3.71	True	4	6561	5338	763
25	1438	3.78	False	0	8748	7310	1094
26	1439	3.93	True	6	13122	11683	1475
27	2879	3.72	True	6	19683	16804	2058
28	3767	3.74	True	4	26244	22477	2878
29	4283	3.81	True	3	39366	35083	3929
30	6299	3.77	True	3	59049	52750	5493
31	10079	3.69	True	7	78732	68653	7669
32	11807	3.75	True	4	118098	106291	10501
33	15287	3.76	True	4	177147	161860	14707
34	21599	3.74	True	6	236196	214597	20476
35	33599	3.69	True	5	354294	320695	28226
36	45197	3.69	True	3	531441	486244	39287
37	56039	3.72	True	5	708588	652549	54817
38	81647	3.69	True	4	1062882	981235	75619
39	98999	3.72	True	5	1594323	1495324	105584
40	163259	3.66	True	3	2125764	1962505	146910
41	203999	3.68	True	5	3188646	2984647	203294
42	241883	3.72	True	3	4782969	4541086	283764
43	371447	3.68	True	4	6377292	6005845	394437
44	540539	3.66	True	3	9565938	9025399	547485
45	590399	3.72	True	6	14348907	13758508	763821
46	907199	3.68	True	7	19131876	18224677	1061367
47	1081079	3.72	True	5	28697814	27616735	1476067
48	1851119	3.65	True	6	43046721	41195602	2057708
49	2041199	3.71	True	7	57395628	55354429	2861449
50	3243239	3.66	True	5	86093442	82850203	3982054
51	3840479	3.70	True	7	129140163	125299684	5552628
52	6562079	3.64	True	7	172186884	165624805	7721319
53	8206559	3.66	True	6	258280326	250073767	10758388
54	11696759	3.65	True	5	387420489	375723730	14994291
55	14648759	3.66	True	5	516560652	501911893	20866891
56	22312799	3.64	True	6	774840978	752528179	29079672
57	27494879	3.66	True	5	1162261467	1134766588	40534895
58	41746319	3.63	True	7	1549681956	1507935637	56439467
59	52252199	3.65	True	5	2324522934	2272270735	78684930
60	78331679	3.63	True	7	3486784401	3408452722	109675955
61	108606959	3.62	True	7	4649045868	4540438909	152788554
62	142990559	3.63	True	6	6973568802	6830578243	213072724
63	203098319	3.62	True	6	10460353203	10257254884	297002458
64	273985919	3.62	True	6	13947137604	13673151685	413944635
65	382021919	3.61	True	7	20920706406	20538684487	577354385
66	495437039	3.62	True	7	31381059609	30885622570	804919055
67	681327359	3.62	True	8	41841412812	41160085453	1122274894
68	1006290359	3.60	True	5	62762119218	61755828859	1565492145
69	1406394359	3.60	True	5	94143178827	92736784468	2182968270
70	1857794399	3.60	True	7	125524238436	123666444037	3044509482
71	2728424159	3.59	True	7	188286357654	185557933495	4247469161
72	3743197919	3.59	True	7	282429536481	278686338562	5923889964
73	5008227839	3.59	True	8	376572715308	371564487469	8263996299
74	6872690159	3.59	True	7	564859072962	557986382803	11530495182
75	9839491199	3.58	True	9	847288609443	837449118244	16084845369
76	13485479039	3.58	True	6	1129718145924	1116232666885	-
77	16724776319	3.59	True	9	1694577218886	1677852442567	-
78	24679458719	3.58	True	7	2541865828329	2517186369610	-
79	35524698479	3.57	True	6	3389154437772	3353629739293	-
80	44211625919	3.59	True	7	5083731656658	5039520030739	-
81	62391692159	3.58	True	8	7625597484987	7563205792828	-
82	93753213119	3.57	True	7	10167463313316	10073710100197	-
83	121551917759	3.57	True	7	15251194969974	15129643052215	-
84	163539961199	3.57	True	7	22876792454961	22713252493762	-
85	250585241759	3.56	True	7	30502389939948	30251804698189	-
86	320429329919	3.57	True	8	45753584909922	45433155580003	-
87	424847520719	3.57	True	7	68630377364883	68205529844164	-
88	630371064959	3.56	True	8	91507169819844	90876798754885	-
89	872573642639	3.56	True	7	137260754729766	136388181087127	-

Concerning the best expressions

There have been conjectures as to what can be the best expression of a number. Notably, it was conjectured, that for prime numbers

$\left\|p\right\| = 1+\left\|p-1\right\|$

and

$\left\|2p\right\| = \min{\{2+\left\|p\right\|, 1+\left\|2p-1\right\|\}}$.

Even though the intuition behind the hypotheses is very natural - if the number is written as a sum, one of the addends should be very small, i.e. 1, since "expected contribution" to the other addend (as an addend to one of its multipliers) is so much more greater - both of these hypotheses have been shown to be false. However, the smallest offending numbers are quite large: 353942783 and 5139300347, respectively.

Some more counterexamples are listed in this table:

$n$	A189124(n)	PrimeQ[A189124(n)]
1	353942783	True
2	516743639	False
3	1163385647	True
4	1542243239	False
5	1932319583	True
6	2336924879	True
7	3113713259	False
8	3444631199	False
9	3878989487	False
10	4103787551	False
11	4166809919	True
12	4937621453	True
13	5123340683	True
14	5170931639	False
15	5184740299	True
16	5200683263	False
17	5390865059	True
18	5455982879	True
19	5467766947	True
20	5570566315	False
21	5876676427	False
22	6020880739	False
23	6213081067	False
24	6432033887	True
25	6459553799	True
26	6545574839	True
27	6714582263	True
28	6888368878	False
29	6988649399	True
30	7349349419	False
31	7354261907	False
32	7378517519	True
33	7515851039	True
34	7657182539	True
35	7756383347	True
36	8219266919	True
37	8265240899	False
38	8267366687	False
39	8312800997	True
40	8319180029	False
41	9299744395	False
42	9307738439	False
43	9312441947	True
44	9417418919	True
45	9649914763	False
46	9687112847	True
47	9796592617	False
48	9797579279	True
49	9810679247	True
50	9816022007	False
51	9819417887	True
52	10009162939	False
53	10251400499	True
54	10278600694	False
55	10752114599	False
56	10795928723	True
57	10858961203	False
58	10870190159	False
59	10900918439	False
60	10948216919	False
61	10948217573	False
62	10982945399	True
63	11030276879	True
64	11430539819	True
65	11520231839	False
66	11879689919	True
67	12279593759	True
68	12402781733	True
69	12440572891	True
70	12464418523	False
71	12483890999	True
72	12506394959	False
73	12571726823	True
74	12580039259	True
75	12686036183	True

Up to $5*10^{11}$ there are 283 counterexamples (all in File:2pCounter.txt) to the second conjecture. The first 50 are:

$n$	$p_n$
1	5139300347
2	7748861399
3	8524048067
4	10363463819
5	10880201627
6	12949888751
7	15664684031
8	16031393999
9	16400847059
10	16434547271
11	16548072767
12	22778889929
13	23253670079
14	23269613327
15	24775428599
16	42372443447
17	44497645811
18	46715524127
19	46748937023
20	49754835719
21	55875353867
22	56525374667
23	57386604347
24	58328815679
25	58436920799
26	61055111087
27	69807954239
28	70008583247
29	70527271499
30	70560682559
31	72319015799
32	72327795389
33	72898696979
34	74595273317
35	74643104087
36	79748569367
37	80770023053
38	85242667319
39	85268381039
40	85892637167
41	85893275771
42	93059836411
43	93301642367
44	94038221177
45	99216675647
46	104618179871
47	110003769791
48	111591768599
49	111835854863
50	112697113391

Programs

A program to calculate complexity in base {1,+,-,*} is found here: File:Minus.txt.