OFFSET
1,2
COMMENTS
The complexity of an integer n is the least number of 1's needed to represent it using only additions, multiplications and parentheses. This does not allow juxtaposition of 1's to form larger integers, so for example, 2 = 1+1 has complexity 2, but 11 does not ("pasting together" two 1's is not an allowed operation).
The complexity of a number has been defined in several different ways by different authors. See the Index to the OEIS for other definitions.
Guy asks if a(p) = a(p-1) + 1 for prime p. Martin Fuller found the least counterexample p = 353942783 in 2008, see Fuller link. - Charles R Greathouse IV, Oct 04 2012
It appears that this sequence is lower than A348262 {1,+,^} only a finite number of times. - Gordon Hamilton and Brad Ballinger, May 23 2022
The second Altman links proves that {a(n) - 3*log_3(n)} is a well-ordered subset of the reals whose intersection with [0,k) has order type omega^k for each positive integer k, so this set itself has order type omega^omega. - Jianing Song, Apr 13 2024
REFERENCES
M. Criton, "Les uns de Germain", Problem No. 4, pp. 13 ; 68 in '7 x 7 Enigmes Et Défis Mathématiques pour tous', vol. 25, Editions POLE, Paris 2001.
R. K. Guy, Unsolved Problems in Number Theory, Sect. F26.
K. Mahler and J. Popken, Over een Maximumprobleem uit de Rekenkunde (Dutch: On a maximum problem in arithmetic), Nieuw Arch. Wiskunde, (3) 1 (1953), pp. 1-15.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
M. F. Hasler, Table of n, a(n) for n = 1..10000
Harry Altman, Highest few sums and products of ones, Preprint, undated but before 2013.
Harry Altman, Integer Complexity and Well-Ordering, arXiv:1310.2894v1 [math.NT] , Oct 10, 2013.
Harry Altman and Joshua Zelinsky, Numbers with integer complexity close to the lower bound, arXiv:1207.4841 [math.NT], 2012.
J. Arias de Reyna, Complejidad de los números naturales, Gaceta R. Soc. Mat. Esp., 3 (2000), 230-250.
J. Arias de Reyna, Complejidad de los números naturales, Gaceta R. Soc. Mat. Esp., 3 (2000), 230-250. [Cached copy, with permission]
J. Arias de Reyna, Complexity of natural numbers, arXiv:2111.03345 [math.NT], 2021.
J. Arias de Reyna and J. van de Lune, The question "How many 1's are needed?" revisited, arXiv preprint arXiv:1404.1850 [math.NT], 2014.
J. Arias de Reyna and J. van de Lune, Algorithms for determining integer complexity, arXiv preprint arXiv:1404.2183 [math.NT], 2014.
Juris Cernenoks, Janis Iraids, Martins Opmanis, Rihards Opmanis and Karlis Podnieks, Integer Complexity: Experimental and Analytical Results II, arxiv:1409.0446 [math.NT], 2014.
Katherine Cordwell, Alyssa Epstein, Anand Hemmady, Steven J. Miller, Eyvindur A. Palsson, Aaditya Sharma, Stefan Steinerberger and Yen Nhi Truong Vu, On algorithms to calculate integer complexity, arXiv:1706.08424 [math.NT], 2017.
Martin N. Fuller, C program
R. K. Guy, Some suspiciously simple sequences, Amer. Math. Monthly 93 (1986), 186-190; 94 (1987), 965; 96 (1989), 905.
Qizheng He, Improved Algorithms for Integer Complexity, arXiv:2308.10301 [cs.DS], 2023.
Janis Iraids, Online calculator of a(n) for n < 10^12
Jānis Iraids, Kaspars Balodis, Juris Čerņenoks, Mārtiņš Opmanis, Rihards Opmanis, and Kārlis Podnieks, Integer complexity: experimental and analytical results, arXiv:1203.6462 [math.NT], 2012.
Daniel A. Rawsthorne, How many 1's are needed?, Fibonacci Quarterly 27 (1989), pp. 14-17.
Srinivas, Vivek V. and Shankar, B. R., Integer Complexity: Breaking the Theta(n^2) barrier, World Academy of Science 41 (2008), pp. 690-691.
Stefan Steinerberger, A Short Note on Integer Complexity, Contributions to Discrete Mathematics, Vol. 9.1 (2014).
Stefan Steinerberger, A short note on integer complexity, Contributions to Discrete Mathematics, Vol. 9.1 (2014). [Cached copy, with permission.]
Joshua Zelinsky, Upper Bounds on Integer Complexity, arXiv preprint (2022). arXiv:2211.02995 [math.NT]
Venecia Wang, A counterexample to the prime conjecture of expressing numbers using just ones, YouTube video, 2012.
Venecia Wang, A counterexample to the prime conjecture of expressing numbers using just ones, Journal of Number Theory, Volume 133, Issue 2, February 2013, Pages 391-397.
Eric Weisstein's World of Mathematics, Integer Complexity
Dimitri Zucker, The Most Underrated Concept in Number Theory, Combo Class Youtube video that mentions this sequence.
FORMULA
It is known that a(n) <= A061373(n) but I think 0 <= A061373(n) - a(n) <= 1 also holds. - Benoit Cloitre, Nov 23 2003 [That's false: the numbers {46, 235, 649, 1081, 7849, 31669, 61993} require {1, 2, 3, 4, 5, 6, 7} fewer 1's in A005245 than in A061373. - Ed Pegg Jr, Apr 13 2004]
It is known from the work of Selfridge and Coppersmith that 3 log_3 n <= a(n) <= 3 log_2 n = 4.754... log_3 n for all n > 1. [Guy, Unsolved Problems in Number Theory, Sect. F26.] - Charles R Greathouse IV, Apr 19 2012 [Comment revised by N. J. A. Sloane, Jul 17 2016]
Zelinsky (2022) improves the upper bound to a(n) <= A*log n where A = 41/log(55296) = 3.754422.... This compares to the constant 2.7307176... of the lower bound. - Charles R Greathouse IV, Dec 11 2022
a(n) >= A007600(n) is a very accurate lower bound. - Mehmet Sarraç, Dec 18 2022
EXAMPLE
From Lekraj Beedassy, Jul 04 2009: (Start)
n.........minimal expression........ a(n) = number of 1's
1..................1...................1
2.................1+1..................2
3................1+1+1.................3
4.............(1+1)*(1+1)..............4
5............(1+1)*(1+1)+1.............5
6............(1+1)*(1+1+1).............5
7...........(1+1)*(1+1+1)+1............6
8..........(1+1)*(1+1)*(1+1)...........6
9...........(1+1+1)*(1+1+1)............6
10..........(1+1+1)*(1+1+1)+1...........7
11.........(1+1+1)*(1+1+1)+1+1..........8
12.........(1+1)*(1+1)*(1+1+1)..........7
13........(1+1)*(1+1)*(1+1+1)+1.........8
14.......{(1+1)*(1+1+1)+1}*(1+1)........8
15.......{(1+1)*(1+1)+1}*(1+1+1)........8
16.......(1+1)*(1+1)*(1+1)*(1+1)........8
17......(1+1)*(1+1)*(1+1)*(1+1)+1.......9
18........(1+1)*(1+1+1)*(1+1+1).........8
19.......(1+1)*(1+1+1)*(1+1+1)+1........9
20......{(1+1+1)*(1+1+1)+1}*(1+1).......9
21......{(1+1)*(1+1+1)+1}*(1+1+1).......9
22.....{(1+1)*(1+1+1)+1}*(1+1+1)+1.....10
23....{(1+1)*(1+1+1)+1}*(1+1+1)+1+1....11
24......(1+1)*(1+1)*(1+1)*(1+1+1).......9
25.....(1+1)*(1+1)*(1+1)*(1+1+1)+1.....10
26....{(1+1)*(1+1)*(1+1+1)+1}*(1+1)....10
27.......(1+1+1)*(1+1+1)*(1+1+1)........9
28......(1+1+1)*(1+1+1)*(1+1+1)+1......10
29.....(1+1+1)*(1+1+1)*(1+1+1)+1+1.....11
30.....{(1+1+1)*(1+1+1)+1}*(1+1+1).....10
31....{(1+1+1)*(1+1+1)+1}*(1+1+1)+1....11
32....(1+1)*(1+1)*(1+1)*(1+1)*(1+1)....10
33...(1+1)*(1+1)*(1+1)*(1+1)*(1+1)+1...11
34..{(1+1)*(1+1)*(1+1)*(1+1)+1}*(1+1)..11
.........................................
(End)
MAPLE
with(numtheory):
a:= proc(n) option remember;
`if`(n=1, 1, min(seq(a(i)+a(n-i), i=1..n/2),
seq(a(d)+a(n/d), d=divisors(n) minus {1, n})))
end:
seq(a(n), n=1..100); # Alois P. Heinz, Apr 18 2012
MATHEMATICA
a[n_] := a[n] = If[n == 1, 1,
Min[Table[a[i] + a[n-i], {i, 1, n/2}] ~Join~
Table[a[d] + a[n/d], {d, Divisors[n] ~Complement~ {1, n}}]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Dec 12 2013, after Alois P. Heinz *)
PROG
(PARI) A005245(n /* start by calling this with the largest needed n */, lim/* see below */) = { local(d); n<6 && return(n);
A005245=vector(n) /* create vector if needed - should better reuse existing data if available */);
A005245[n]=min( vecmin(vector(min(n\2, if(lim>0, lim, n)), k, A005245[k]+A005245[n-k])) /* additive possibilities - if lim>0 is given, consider a(k)+a(n-k) only for k<=lim - we know it is save to use lim=1 up to n=2e7 */, if( isprime(n), n, vecmin(vector((-1+#d=divisors(n))\2, i, A005245[d[i+1]]+A005245[d[ #d-i]]))/* multiplicative possibilities */))}
\\ See also the Python program by Tim Peters at A005421.
\\ M. F. Hasler, Jan 30 2008
(Haskell)
import Data.List (genericIndex)
a005245 n = a005245_list `genericIndex` (n-1)
a005245_list = 1 : f 2 [1] where
f x ys = y : f (x + 1) (y : ys) where
y = minimum $
(zipWith (+) (take (x `div` 2) ys) (reverse ys)) ++
(zipWith (+) (map a005245 $ tail $ a161906_row x)
(map a005245 $ reverse $ init $ a161908_row x))
-- Reinhard Zumkeller, Mar 08 2013
(Python)
from functools import lru_cache
from itertools import takewhile
from sympy import divisors
@lru_cache(maxsize=None)
CROSSREFS
KEYWORD
AUTHOR
EXTENSIONS
More terms from David W. Wilson, May 15 1997
STATUS
approved