A025280 -id:A025280 - cp's OEIS frontend

A005245 The (Mahler-Popken) complexity of n: minimal number of 1's required to build n using + and *.

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

Offset: 1

N. J. A. Sloane

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

			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)

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).

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.
Index to sequences related to the complexity of n

Cf. A025280 (variant using +, *, and ^), A091333 (using +, -, and *), A091334 (using +, -, *, and ^), A348089 (using +, -, *, / and ^), A348262 (using + and ^).

Cf. A000792 (largest integer of given complexity), A003313, A076142, A076091, A061373, A005421, A064097, A005520, A003037, A161906, A161908, A244743.

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

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

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 *)

A005245(n /* start by calling this with the largest needed n */, lim/* see below */) = { local(d); n<6 && return(n);
if(n<=#A005245, A005245[n]&return(A005245[n]) /* return memoized result if available */,
A005245=vector(n) /* create vector if needed - should better reuse existing data if available */);
for(i=1, n-1, A005245[i] || A005245[i]=A005245(i,lim)); /* compute all previous elements */
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

from functools import lru_cache
from itertools import takewhile
from sympy import divisors
@lru_cache(maxsize=None)
def A005245(n): return min(min(A005245(a)+A005245(n-a) for a in range(1,(n>>1)+1)),min((A005245(d)+A005245(n//d) for d in takewhile(lambda d:d*d<=n,divisors(n)) if d>1),default=n)) if n>1 else 1 # Chai Wah Wu, Apr 29 2023

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

More terms from David W. Wilson, May 15 1997

A005520 Smallest number of complexity n: smallest number requiring n 1's to build using + and *.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 10, 11, 17, 22, 23, 41, 47, 59, 89, 107, 167, 179, 263, 347, 467, 683, 719, 1223, 1438, 1439, 2879, 3767, 4283, 6299, 10079, 11807, 15287, 21599, 33599, 45197, 56039, 81647, 98999, 163259, 203999, 241883, 371447, 540539, 590399, 907199

Offset: 1

N. J. A. Sloane

Ed Pegg Jr, www.mathpuzzle.com, Apr 10 2001, notes that all the new terms are -1 mod 120. - That is, this holds at least for all terms from a(45) = 590399 up to the highest known term a(89) given in the b-file at the end of 2015. Comment clarified by Antti Karttunen, Dec 14 2015
Largest number of complexity n is given by A000792. - David W. Wilson, Oct 03 2005
After 1438 = 2 * 719, all elements through 8206559 are primes. Equivalently, except for a(4) = 4, a(7) = 10, a(10) = 22 and a(25) = 1438, we have a(1) through a(53) are all primes. - Jonathan Vos Post, Apr 07 2006
a(54)-a(89) are all primes. - Janis Iraids, Apr 21 2011
Previous observations (primes with property -1 mod 120) still hold. - Martins Opmanis, Oct 16 2009
The prime 353942783 = A189125(1) = 2*3 + (1 + 2^2*3^2)*(2 + 3^4(1 + 2*3^10)) is the smallest counterexample to Guy's first hypothesis on integer complexity. - Jonathan Vos Post, Mar 30 2012
The sequence A265360 (second smallest number of complexity n) seems to have similar properties. - Janis Iraids via Antti Karttunen, Dec 15 2015

			Examples from Piotr Fabian:
1 = 1, 1 "one": first 1, a(1) = 1
2 = 1+1, 2 "ones": first 2, a(2) = 2
3 = 1+1+1, 3 "ones": first 3, a(3) = 3
4 = 1+1+1+1, 4 "ones": first 4, a(4) = 4
5 = 1+1+1+1+1, 5 "ones": first 5, a(5) = 5
6 = (1+1)*(1+1+1), 5 "ones"
7 = 1+((1+1)*(1+1+1)), 6 "ones": first 6, a(6) = 7
8 = (1+1)*(1+1+1+1), 6 "ones"
9 = (1+1+1)*(1+1+1), 6 "ones"
10 = 1+((1+1+1)*(1+1+1)), 7 "ones": first 7, a(7) = 10
11 = 1+(1+(1+1+1)*(1+1+1)), 8 "ones": first 8, a(8) = 11
12 = (1+1)*((1+1)*(1+1+1)), 7 "ones"

R. K. Guy, Unsolved Problems in Number Theory, Sec. F26 (related material).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Janis Iraids, Table of n, a(n) for n = 1..89
Kazuyuki Amano, Integer Complexity and Mixed Binary-Ternary Representation, 33rd Int'l Symp. Alg. Comp. (ISAAC 2022) Leibniz Int'l Proc. Informatics (LIPIcs) Vol. 248.
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. See also Integers (2024) Vol. 24, A19, p. 6.
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.
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.
Ed Pegg Jr., Math Games: Integer Complexity, www.mathpuzzle.com, April 12, 2004.
D. A. Rawsthorne, How many 1's are needed?, Fib. Quart. 27 (1989), 14-17.
Eric Weisstein's World of Mathematics, Integer Complexity
Index to sequences related to the complexity of n

Cf. A005245, A025280, A003037, A005421, A048183, A181898, A182061, A265360.

N = 10^6;
fact = cell(1,N);
for n = 2:sqrt(N)
  for m = [n^2:n:N]
    fact{m} = [fact{m},n];
  end
end
R = zeros(1,N);
R(1) = 1;
A(1) = 1;
mmax = 1;
for n = 2:N
  m = min(R(1:floor(n/2)) + R([n-1:-1:ceil(n/2)]));
  if numel(fact{n}) > 0
    m = min(m, min(R(fact{n}) + R(n ./ fact{n})));
  end
  R(n) = m;
  if m > mmax
    A(m) = n;
    mmax = m;
  elseif A(m) == 0
    A(m) = n;
  end
end
A % Robert Israel, Dec 14 2015

N:= 100000: # to get all terms <= N
R:= Vector(N):
R[1]:= 1: A[1]:= 1:
for n from 2 to N do
  inds:= [seq(n-i,i=1..floor(n/2))];
  m:= min(R[1..floor(n/2)] + R[inds]);
  for d in select(`<=`,numtheory:-divisors(n),floor(sqrt(n))) minus {1} do
    m:= min(m, R[d]+R[n/d])
  od;
  R[n]:= m;
  if not assigned(A[m]) then A[m]:= n fi;
od:
seq(A[m],m=1..max(R)); # Robert Israel, Dec 14 2015

nn = 10000;
R = Table[0, nn];
R[[1]] = 1; Clear[A]; A[1] = 1;
For[n = 2, n <= nn, n++,
  inds = Table[n-i, {i, 1, n/2}];
  m = Min[R[[1 ;; Floor[n/2]]] + R[[inds]]];
  Do[
    m = Min[m, R[[d]] + R[[n/d]]], {d,
    Select[Rest[Divisors[n]], # <= Sqrt[n]&]}
  ];
  R[[n]] = m;
  If[!IntegerQ[A[m]], A[m] = n];
];
Table[A[m], {m, 1, Max[R]}] (* Jean-François Alcover, Aug 05 2018, after Robert Israel *)

def aupton(nn):
  alst, R = [1], {0: {1}} # R[n] is set reachable using n+1 1's (n ops)
  for n in range(1, nn):
    R[n]  = set(a+b for i in range(n//2+1) for a in R[i] for b in R[n-1-i])
    R[n] |= set(a*b for i in range(n//2+1) for a in R[i] for b in R[n-1-i])
    alst.append(min(R[n] - R[n-1]))
  return alst
print(aupton(35)) # Michael S. Branicky, Jun 08 2021

Corrected and extended by David W. Wilson, May 1997
Extended to terms a(40)=163259 and a(41)=203999 by John W. Layman, Nov 03 1999
Further terms from Piotr Fabian (PCF(AT)who.net), Mar 30 2001
a(68)-a(89) from Janis Iraids, Apr 20 2011

A091333 Number of 1's required to build n using +, -, *, and parentheses.

Original entry on oeis.org

1, 2, 3, 4, 5, 5, 6, 6, 6, 7, 8, 7, 8, 8, 8, 8, 9, 8, 9, 9, 9, 10, 10, 9, 10, 10, 9, 10, 11, 10, 11, 10, 11, 11, 11, 10, 11, 11, 11, 11, 12, 11, 12, 12, 11, 12, 12, 11, 12, 12, 12, 12, 12, 11, 12, 12, 12, 13, 13, 12, 13, 13, 12, 12, 13, 13, 14, 13, 13, 13, 13, 12, 13, 13, 13, 13, 14

Offset: 1

Jens Voß, Dec 30 2003

nonn

Consider an alternate complexity measure b(n) which gives the minimum number of 1's necessary to build n using +, -, *, and / (where this additional operation is strict integer division, defined only for n/d where d|n). It turns out that b(n) coincides with a(n) for all n up to 50221174, see A348069. - Glen Whitney, Sep 23 2021

In respect of the previous comment: when creating A362471, where repunits are allowed, we found a difference if we allowed n/d with noninteger (intermediate) results. So, see also A362626. - Peter Munn, Apr 29 2023

			A091333(23) = 10 because 23 = (1+1+1+1) * (1+1+1) * (1+1) - 1. (Note that 23 is also the smallest index at which A091333 differs from A005245.)

Janis Iraids, Table of n, a(n) for n = 1..10000
Juris Cernenoks, Janis Iraids, Martins Opmanis, Rihards Opmanis and Karlis Podnieks, Integer Complexity: Experimental and Analytical Results II, arXiv:1409.0446 [math.NT], 2014.
Janis Iraids, A C program to compute the sequence
J. Iraids, K. Balodis, J. Cernenoks, M. Opmanis, R. Opmanis and K. Podnieks, Integer Complexity: Experimental and Analytical Results. arXiv preprint arXiv:1203.6462 [math.NT], 2012. - From _N. J. A. Sloane_, Sep 22 2012
Index to sequences related to the complexity of n

Cf. A005245 (variant using + and *), A025280 (using +, *, and ^), A091334 (using +, -, *, and ^), A348089 (using +, -, *, /, and ^), A348262 (using + and ^).

Cf. A348069, A362471, A362626.

from functools import cache
@cache
def f(m):
    if m == 0: return set()
    if m == 1: return {1}
    out = set()
    for j in range(1, m//2+1):
        for x in f(j):
            for y in f(m-j):
                out.update([x + y, x * y])
                if x != y: out.add(abs(x-y))
    return out
def aupton(terms):
    tocover, alst, n = set(range(1, terms+1)), [0 for i in range(terms)], 1
    while len(tocover) > 0:
        for k in f(n) - f(n-1):
            if k <= terms:
                alst[k-1] = n
                tocover.discard(k)
        n += 1
    return alst
print(aupton(77)) # Michael S. Branicky, Sep 28 2021

A003037 Smallest number of complexity n: smallest number requiring n 1's to build using +, * and ^.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 11, 13, 21, 23, 41, 43, 71, 94, 139, 211, 215, 431, 863, 1437, 1868, 2855, 5737, 8935, 15838, 15839, 54357, 95597, 139117, 233195, 470399, 1228247, 2183791, 4388063, 6945587, 13431919, 32329439, 46551023

Offset: 1

N. J. A. Sloane

The complexity of an integer n is the least number of 1's needed to represent it using only additions, multiplications, exponentiation 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 (concatenating 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. - Jonathan Vos Post, Oct 20 2007

			An example (usually nonunique) of the derivation of the first 10 values.
a(1) = 1, the number of 1's in "1."
a(2) = 2, the number of 1's in "1+1 = 2."
a(3) = 3, the number of 1's in "1+1+1 = 3."
a(4) = 4, the number of 1's in "1+1+1+1 = 4."
a(5) = 5, the number of 1's in "1+1+1+1+1 = 5."
a(6) = 7, since there are 6 1's in "((1+1)*(1+1+1))+1 = 7."
a(7) = 11, since there are 7 1's in "((1+1+1)^(1+1))+1+1 = 11."
a(8) = 13, since there are 8 1's in "((1+1+1)*(1+1+1+1))+1 = 13."
a(9) = 21, since there are 9 1's in "(1+1+1)*(((1+1)*(1+1+1))+1) = 21."
a(10) = 23, since there are 10 1's in "1+((1+1)*(((1+1+1)^(1+1))+1+1)) = 23."

W. A. Beyer, M. L. Stein and S. M. Ulam, The Notion of Complexity. Report LA-4822, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, December 1971.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

W. A. Beyer, Letter to N. J. A. Sloane, 1980
W. A. Beyer, M. L. Stein and S. M. Ulam, The Notion of Complexity. Report LA-4822, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, December 1971. [Annotated scanned copy]
Index to sequences related to the complexity of n

Cf. A025280, A005520, A005245, A005421, A117618.

xmax:= 5:  # get terms <= 10^xmax
C[1]:= {1}: A[1]:= 1: CU[1]:= {1}:
for n from 2 do
   C[n]:= {seq(seq(seq(op(select(`<=`,
[a+b,a*b,`if`(b*ilog10(a) <= xmax,a^b,NULL),`if`(a*ilog10(b) <= xmax,b^a,NULL)]
         ,10^xmax)),b=C[n-k]),a=C[k]),k=1..floor(n/2))}
         minus CU[n-1];
   if C[n] = {} then break fi;
   A[n]:= min(C[n]);
   CU[n]:= CU[n-1] union C[n];
od:
seq(A[i],i=1..n-1); # Robert Israel, Jan 08 2015

More terms from David W. Wilson, May 15 1997

More terms from Sean A. Irvine, Jan 07 2015

A091334 Number of 1's required to build n using +, -, *, ^, and parentheses.

Original entry on oeis.org

1, 2, 3, 4, 5, 5, 6, 5, 5, 6, 7, 7, 8, 8, 7, 6, 7, 7, 8, 8, 9, 9, 9, 8, 7, 7, 6, 7, 8, 9, 8, 7, 8, 9, 8, 7, 8, 9, 10, 10, 11, 11, 12, 11, 10, 11, 10, 9, 8, 9, 10, 9, 9, 8, 9, 9, 10, 10, 11, 11, 10, 9, 8, 7, 8, 9, 10, 11, 11, 10, 10, 9, 10, 10, 10, 11, 11, 10, 9, 8, 7, 8, 9, 10, 11, 12, 11, 12, 12

Offset: 1

Jens Voß, Dec 30 2003

nonn

One strategy for computing integer complexities like a(n) is to proceed in rounds, successively determining all n for which a(n) = 2, 3, 4, 5, and so on. In each round one takes all operations of pairs of numbers whose already known lesser complexities sum to the current value. The difficulty with this particular a(n) is that exponentiation quickly produces very large values, which conceivably could be near each other and yield a value of a(n) for small n by taking their difference in a later round. However, one can safely compute small values up to 19 by ignoring intermediate results larger than 2^65: this doesn't ignore anything except 2^81, 2^256, and 2^512 in the ninth round, and those values will not become involved in differences that could affect arguments n less than 2^65 until at least round 19, so all small values up to 19 produced in this way will be correct. Doing so gives values for all n from a(1)=1 up to a(3305)=19. It is possible based on the above that a(3306) might be 19, if some large result from round 10 differs from one of the three ignored values in round 9 by exactly 3306. That just about surely doesn't happen, but more careful reasoning would be needed to prove the correct value a(3306) = 20. - Glen Whitney, Sep 22 2021

			A091334(15) = 7 because 15 = (1+1+1+1)^(1+1) - 1. (Note that 15 is also the smallest index at which A091334 differs from A025280.)

Glen Whitney, Table of n, a(n) for n = 1..3305
J. Iraids, K. Balodis, J. Cernenoks, M. Opmanis, R. Opmanis and K. Podnieks, Integer Complexity: Experimental and Analytical Results. arXiv preprint arXiv:1203.6462, 2012. - From _N. J. A. Sloane_, Sep 22 2012
Index to sequences related to the complexity of n
Glen Whitney, Python3.8 program computing up to a(3305)

Cf. A005245 (variant using + and *), A025280 (using +, *, and ^), A091333 (using +, -, and *), A348089 (using +, -, *, /, and ^), A348262 (using + and ^).

A182002 Smallest positive integer that cannot be computed using exactly n n's, the four basic arithmetic operations (+, -, *, /), and the parentheses.

Original entry on oeis.org

2, 2, 1, 10, 13, 22, 38, 91, 195, 443, 634, 1121, 3448, 6793, 17692

Offset: 1

Ali Dasdan, Apr 05 2012

			a(2) = 2 because two 2's can produce 0 = 2-2, 1 = 2/2, 4 = 2+2 = 2*2, so the smallest positive integer that cannot be computed is 2.
a(3) = 1 because no expression with three 3's gives 1.

Cf. A005245, A005520, A003313, A076142, A076091, A061373, A005421, A064097, A025280, A003037, A117618.

Cf. A171826, A171827, A171828, A171829, A258068, A258069, A258070, A258071.

f:= proc(n,b) option remember;
      `if`(n=1, {b}, {seq(seq(seq([k+m, k-m, k*m,
      `if`(m=0, NULL, k/m)][], m=f(n-i, b)), k=f(i, b)), i=1..n-1)})
    end:
a:= proc(n) local i, l;
      l:= sort([infinity, select(x-> is(x, integer) and x>0, f(n, n))[]]);
      for i do if l[i]<>i then return i fi od
    end:
seq(a(n), n=1..8); # Alois P. Heinz, Apr 13 2012

from fractions import Fraction
from functools import lru_cache
def a(n):
    @lru_cache()
    def f(m):
        if m == 1: return {Fraction(n, 1)}
        out = set()
        for j in range(1, m//2+1):
            for x in f(j):
                for y in f(m-j):
                    out.update([x + y, x - y, y - x, x * y])
                    if y: out.add(Fraction(x, y))
                    if x: out.add(Fraction(y, x))
        return out
    k, s = 1, f(n)
    while k in s: k += 1
    return k
print([a(n) for n in range(1, 10)]) # Michael S. Branicky, Jul 29 2022

a(11)-a(12) from Alois P. Heinz, Apr 22 2012
a(13)-a(14) from Michael S. Branicky, Jul 29 2022
a(15) from Michael S. Branicky, Jul 27 2023

A348089 Number of 1's required to build n using +, -, *, /, and ^.

Original entry on oeis.org

1, 2, 3, 4, 5, 5, 6, 5, 5, 6, 7, 7, 8, 8, 7, 6, 7, 7, 8, 8, 9, 9, 9, 8, 7, 7, 6, 7, 8, 9, 8, 7, 8, 9, 8, 7, 8, 9, 10, 10, 10, 11, 12, 11, 10, 11, 10, 9, 8, 9, 10, 9, 9, 8, 9, 9, 10, 10, 11, 11, 10, 9, 8, 7, 8, 9, 10, 11, 11, 10, 10, 9, 10, 10, 10

Offset: 1

Glen Whitney, Sep 28 2021

nonn

Here division is taken to be strict integer division, i.e., j/k is defined only if k|j.

Because of the presence of exponentiation, division reduces the value of a(n) as compared with A091334(n) (which allows the same operations except division) far more often than adding division to A091333 does; see A348069.

			Because 41 = ((1+1+1)^(1+1+1+1) + 1)/(1+1), and there is no expression with these operators and fewer ones that evaluates to 41, a(41) = 10. Note that 41 is the least n such that a(n) < A091334(n).

Glen Whitney, Table of n, a(n) for n = 1..3802
Glen Whitney, Python3.8 program to compute a(n) for some values of n

Cf. A005245 (variant using + and *), A025280 (using +, *, and ^), A091333 (using +, -, and *), A091334 (using +, -, *, and ^), A348262 (using + and ^).

Cf. A348069.

A348262 Number of 1's required to build n using + and ^.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 5, 5, 6, 7, 8, 9, 10, 11, 6, 7, 8, 9, 10, 11, 12, 13, 11, 7, 8, 6, 7, 8, 9, 10, 7, 8, 9, 10, 8, 9, 10, 11, 12, 12, 13, 12, 13, 13, 14, 15, 13, 9, 10, 11, 12, 13, 12, 13, 14, 14, 14, 13, 14, 15, 16, 14, 7, 8, 9, 10, 11, 12, 13, 14, 12, 12, 13, 14, 15, 16, 17

Offset: 1

Glen Whitney, Oct 09 2021

nonn

			11+111++^ is a minimal-length RPN formula with value 8, using just these operators. It contains five occurrences of the symbol "1". Hence, a(8) = 5.

Glen Whitney, Table of n, a(n) for n = 1..10000
Edinah K. Ghang and Doron Zeilberger, Zeroless Arithmetic: Representing Integers ONLY using ONE, arXiv:1303.0885 [math.CO], 2013.
Glen Whitney, Python3.8 program to compute a(n)
Index to sequences related to the complexity of n

Cf. A213924 (expression-length complexity with the same set {1,+,^}).

Cf. A005245 (variant using + and *), A025280 (using +, *, and ^), A091333 (using +, -, and *), A091334 (using +, -, *, and ^), A348089 (using +, -, *, /, and ^).

a(n) = (A213924(n) + 1)/2.

A005208 Operator-oriented complexity of n, i.e., the minimum number of occurrences of +, *, and ^ needed to build n from a supply of ones.

Original entry on oeis.org

0, 1, 2, 3, 4, 4, 5, 4, 4, 5, 6, 6, 7, 7, 7, 5, 6, 6, 7, 7, 8, 8, 9, 7, 6, 7, 5, 6, 7, 8, 9, 6, 7, 8, 9, 6, 7, 8, 9, 9, 10, 10, 11, 10, 9, 10, 11, 8, 7, 8, 9, 9, 10, 7, 8, 8, 9, 9, 10, 10, 11, 11, 10, 6, 7, 8, 9, 10, 11, 11, 12, 8, 9, 9, 9, 10, 11, 10, 11, 10

Offset: 1

N. J. A. Sloane

nonn

The formula is correct because k ones require exactly k - 1 binary operators to reduce to a single value. - Glen Whitney, Oct 06 2021

W. A. Beyer, M. L. Stein and S. M. Ulam, The Notion of Complexity. Report LA-4822, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, December 1971.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..10000
W. A. Beyer, Letter to N. J. A. Sloane, 1980
W. A. Beyer, M. L. Stein and S. M. Ulam, The Notion of Complexity. Report LA-4822, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, December 1971. [Annotated scanned copy]
Index to sequences related to the complexity of n

Cf. A025280.

a(n) = A025280(n) - 1.

Name clarified by Glen Whitney, Oct 06 2021

A133344 Complexity of the number n, counting 1's and built using +, *, ^ and # representing concatenation.

Original entry on oeis.org

1, 2, 3, 4, 5, 5, 6, 5, 5, 6, 2, 3, 4, 5, 6, 6, 7, 6, 6, 7, 3, 4, 5, 5, 6, 6, 6, 7, 7, 8, 4, 5, 5, 6, 7, 6, 7, 8, 7, 8, 5, 5, 6, 6, 7, 7, 8, 7, 8, 8, 6, 7, 7, 8, 7, 8, 9, 9, 9, 8, 6, 6, 6, 7, 8, 7, 8, 8, 8, 9, 7, 8, 8, 8, 9, 10, 8, 9, 10, 10, 6, 7, 8, 7, 8, 8

Offset: 1

Jonathan Vos Post, Oct 20 2007

The complexity of an integer n is the least number of 1's needed to represent it using only additions, multiplications, exponentiation and parentheses. This allows juxtaposition of numbers to form larger integers, so for example, 2 = 1+1 has complexity 2, but unlike A003037, so does 11 = 1#1 (concatenating two numbers is an allowed operation). Similarly a(111) = 3. The complexity of a number has been defined in several different ways by different authors. See the Index to the OEIS for other definitions.

			An example (usually nonunique) of the derivation of the first 22 values.
a(1) = 1, the number of 1's in "1."
a(2) = 2, the number of 1's in "1+1 = 2."
a(3) = 3, the number of 1's in "1+1+1 = 3."
a(4) = 4, the number of 1's in "1+1+1+1 = 4."
a(5) = 5, the number of 1's in "1+1+1+1+1 = 5."
a(6) = 5, since there are 5 1's in "((1+1)*(1+1+1)) = 6."
a(7) = 6, since there are 6 1's in "1+(((1+1)*(1+1+1))) = 7."
a(8) = 5, since there are 5 1's in "(1+1)^(1+1+1) = 8."
a(9) = 5, since there are 5 1's in "(1+1+1)^(1+1) = 9."
a(10) = 6 since there are 6 1's in "1+((1+1+1)^(1+1)) = ten.
a(11) = 2 since there are 2 1's in "1#1 = eleven."
a(12) = 3 since there are 3 1's in "1+(1#1) = twelve."
a(13) = 4 since there are 4 1's in "1+1+(1#1) = thirteen."
a(14) = 5 since there are 5 1's in "1+1+1+(1#1) = fourteen."
a(16) = 6 since there are 6 1's in "(1+1+1+1)^(1+1)."
a(17) = 7 since there are 7 1's in "1+((1+1+1+1)^(1+1))."
a(18) = 6 since there are 6 1's in "1#((1+1)^(1+1+1))."
a(19) = 6 since there are 6 1's in "1#((1+1+1)^(1+1))."
a(20) = 7 since there are 7 1's in "(1#1)+((1+1+1)^(1+1))."
a(21) = 3 since there are 3 1's in "(1+1)#1."
a(22) = 4 since 22 = (1+1)*(1#1) = (1#1)+(1#1) = (1+1)#(1+1).

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Index to sequences related to the complexity of n

Cf. A003037, A025280, A005520, A005245, A005421, A117618.

with(numtheory):
a:= proc(n) option remember; local r; `if`(n=1, 1, min(
       seq(a(i)+a(n-i), i=1..n-1),
       seq(a(d)+a(n/d), d=divisors(n) minus {1, n}),
       seq(`if`(cat("", n)[i+1]<>"0", a(iquo(n, 10^(length(n)-i),
           'r'))+a(r), NULL), i=1..length(n)-1),
       seq(a(root(n, p))+a(p), p=divisors(igcd(seq(i[2],
           i=ifactors(n)[2]))) minus {0, 1})))
    end:
seq(a(n), n=1..120);  # Alois P. Heinz, Nov 06 2013

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A005245 The (Mahler-Popken) complexity of n: minimal number of 1's required to build n using + and *.

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A005520 Smallest number of complexity n: smallest number requiring n 1's to build using + and *.

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A091333 Number of 1's required to build n using +, -, *, and parentheses.

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A003037 Smallest number of complexity n: smallest number requiring n 1's to build using +, * and ^.

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A091334 Number of 1's required to build n using +, -, *, ^, and parentheses.

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A182002 Smallest positive integer that cannot be computed using exactly n n's, the four basic arithmetic operations (+, -, *, /), and the parentheses.

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A348089 Number of 1's required to build n using +, -, *, /, and ^.

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