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A005245 Complexity of n: minimal number of 1's required to build n using + and *.
(Formerly M0457)
35
1, 2, 3, 4, 5, 5, 6, 6, 6, 7, 8, 7, 8, 8, 8, 8, 9, 8, 9, 9, 9, 10, 11, 9, 10, 10, 9, 10, 11, 10, 11, 10, 11, 11, 11, 10, 11, 11, 11, 11, 12, 11, 12, 12, 11, 12, 13, 11, 12, 12, 12, 12, 13, 11, 12, 12, 12, 13, 14, 12, 13, 13, 12, 12, 13, 13, 14, 13, 14, 13, 14, 12, 13, 13, 13, 13, 14, 13, 14 (list; graph; refs; listen; history; text; internal format)
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

Altman (2013) writes: "Define |n| to be the complexity of n, the smallest number of ones needed to write n using an arbitrary combination of addition and multiplication. John Selfridge showed that |n| >=3 log_3 n for all n. Define the defect of n, delta(n), to be |n| - 3 log_3 n. In this paper, we consider the set D := {delta(n): n >= 1}. We show that as a subset of the real numbers, the set D is well-ordered, of order type omega^omega. More specifically, for an integer k >=1, Dcap[0,k) has order type omega^k. We also consider some other sets related to D, and show that these too are well-ordered and have order type omega^omega." - Jonathan Vos Post, Oct 10 2013

REFERENCES

M. Criton, "Les uns de Germain", Problem No. 4, pp. 13 ; 68 in '7 x 7 Enigmes Et Defis Mathematiques pour tous', vol. 25, Editions POLE, Paris 2001.

J. Arias de Reyna, Complejidad de los números naturales, Gaceta R. Soc. Mat. Esp., 3 (2000), 230-250.

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).

Stefan Steinerberger, A Short Note on Integer Complexity, Contributions to Discrete Mathematics, Vol. 9.1 (2014); available from http://cdm.journalhosting.ucalgary.ca/cdm/index.php/cdm/article/viewFile/361/181

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. [Cached copy, with permission]

J. Arias de Reyna, 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, 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, Karlis Podnieks, Integer Complexity: Experimental and Analytical Results II, arxiv:1409.0446 [math.NT], 2014.

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.

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). [Cached copy, with permission.]

Venecia Wang, A counterexample to the prime conjecture of expressing numbers using just ones, Journal of Number Theory, submitted. video abstract

Eric Weisstein's World of Mathematics, Integer Complexity

Index to sequences related to the complexity of n

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]

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 [Sequence @@ Table [a[i] + a[n-i], {i, 1, n/2}], Sequence @@ 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, from M. F. Hasler, Jan 30 2008) A005245(n /* start by calling this with the largest needed n */, lim/* see below */) = { local(d); n<6&return(n);

if(n<=#A05245, A05245[n]&return(A05245[n]) /* return memoized result if available */,

A05245=vector(n) /*create vector if needed - should better re-use exiting data if available */);

for(i=1, n-1, A05245[i] || A05245[i]=A005245(i, lim)); /* compute all previous elements */

A05245[n]=min( vecmin(vector(min(n\2, if(lim>0, lim, n)), k, A05245[k]+A05245[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, A05245[d[i+1]]+A05245[d[ #d-i]]))/* multiplicative possibilities */))}

See also the Python program by Tim Peters at A005421.

(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

CROSSREFS

Cf. A000792 (largest integer of given complexity), A003313, A076142, A076091, A061373, A005421, A064097, A005520, A025280, A003037, A161906, A161908.

Sequence in context: A007600 A195872 A091333 * A061373 A104135 A046108

Adjacent sequences:  A005242 A005243 A005244 * A005246 A005247 A005248

KEYWORD

nonn,nice,look,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from David W. Wilson, May 15 1997

STATUS

approved

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Last modified July 29 03:28 EDT 2016. Contains 275161 sequences.