A005245 -id:A005245 - cp's OEIS frontend

A306560 Smallest number of 1's required to build n using +, *, ^ and tetration.

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

Offset: 1

Robin Powell, Feb 23 2019

nonn

			a(16) = 5 because 16 = (1+1)^^(1+1+1). (Note that 16 is also the smallest index at which this sequence differs from A025280.)
a(34) = 8 because 34 = ((1+1)^^(1+1+1)+1)*(1+1). - _Jens Ahlström_, Jan 11 2023

Index to sequences related to the complexity of n

Cf. A005245, A025280, A323727.

from functools import lru_cache
from sympy import factorint, divisors
tetration = {2**2**2: 5, 2**2**2**2: 6, 3**3: 5, 4**4: 6, 5**5: 7}
@lru_cache(maxsize=None)
def a(n):
    res = n
    if n < 6:
        return res
    if n in tetration:
        return tetration[n]
    for i in range(1, n):
        res = min(res, a(i) + a(n-i))
    for d in [i for i in divisors(n) if i not in {1, n}]:
        res = min(res, a(d) + a(n//d))
    factors = factorint(n)
    exponents = set(factors.values())
    if len(exponents) == 1:
        e = exponents.pop()
        if e > 1:
            res = min(res, a(sum(factors.keys())) + a(e))
    return res
# Jens Ahlström, Jan 11 2023

a(34) onward corrected by Jens Ahlström, Jan 11 2023

A319975 Smallest number of complexity n with respect to the operations {1, shift, multiply}.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 10, 11, 14, 19, 22, 23, 38, 43, 58, 59, 89, 107, 134, 167, 179, 263, 347, 383, 537, 713, 719, 1103, 1319, 1439, 2099, 2879, 3833, 4283, 5939, 6299, 9059, 12239, 15118, 19079, 23039, 26459, 44879, 49559, 66239, 78839, 98999, 137339

Offset: 1

N. J. A. Sloane, Oct 11 2018

nonn

The shift operation here is also sometimes called successor, see A263283.

Note this complexity measure counts both operands (the ones) and operators (the shifts and multiplications), whereas most of the complexity measures in the crossrefs count only operands. However, in the presence of successor it would not make sense to count only operands, since any number can be expressed with a single occurrence of 1. - Glen Whitney, Oct 06 2021

			1 = 1 has complexity 1
2 = S1 has complexity 2
3 = SS1 has complexity 3
4 = SSS1 has complexity 4
5 = SSSS1 has complexity 5
6 = SSSSS1 has complexity 6
7 = SSSSSS1 has complexity 7
10 = S*SS1SS1 = shift(product of (3 and 3)) has complexity 8
(Note that 8 = *S1SSS1 has complexity 7)
11 = SS*SS1SS1 has complexity 9
14 = SS*SS1SSS1 has complexity 10

Michael S. Branicky, Table of n, a(n) for n = 1..67
Akshunna Shaurya Dogra, Minimal Representations of Natural Numbers Under a Set of Operators, arXiv preprint 1801.01360 [math.HO], Jan 2018.
Index to sequences related to the complexity of n.

Smallest number of complexity n (other definitions): A003037, A005520, A244743, A259466, and A265360.

Other definitions of the complexity of n: A005208, A005245, A025280, and A099053.

def aupton(nn):
    alst, R, allR = [1], {1: {1}}, {1} # R[n] is set reachable using n ops
    for n in range(2, nn+1):
        R[n]  = set(a+1 for a in R[n-1])
        R[n] |= set(a*b for i in range(1, (n+1)//2) for a in R[i] for b in R[n-1-i])
        alst.append(min(R[n] - allR))
        allR |= R[n]
    return alst
print(aupton(49)) # Michael S. Branicky, Oct 06 2021

A348083 Number of positive numbers that can be built with n ones using +, -, and *, and require at least n ones.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 2, 6, 6, 8, 13, 18, 21, 35, 45, 61, 90, 121, 162, 241, 323, 450, 638, 865, 1233, 1698, 2349, 3315, 4592, 6382, 8970, 12421, 17351, 24320, 33714, 47218, 65978, 91784, 128177, 179807, 249689, 349549, 489341, 681468, 953769, 1334490, 1860641, 2606043, 3643618, 5086481, 7124229, 9960420

Offset: 1

Glen Whitney, Sep 27 2021

nonn

a(n+1)/a(n) appears from the values through n=63 to be oscillating in a narrowing range around 7/5.

			For n=5, there are two numbers whose minimal expression using 1,+,-, and * uses five ones: 5 = 1+1+1+1+1 and 6 = (1+1)*(1+1+1), so a(5) = 2.
For n=10, there are eight numbers whose minimal expression uses ten ones: 22 = 3(2*3+1)+1, 23 = 2*2*2*3-1, 25 = 5*5, 26 = 3*3*3-1, 28 = 3*3*3+1, 30 = 2*3*5, 32 = 2*2*2*2*2, and 36 = 2*2*3*3. We use numbers k=1..5 in these expressions because each takes k ones to express. Note that n=10 is also the least n for which a(n) differs from A005421(n), which counts the solutions to A005245(k) = n.

Glen Whitney, Table of n, a(n) for n = 1..64 (This file produced with the same C code that generated A348069, simply with division eliminated from the loop over operators.)

Cf. A005421, A091333, A005245.

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 list(out)
def a(n): return len(f(n)) - len(f(n-1))
print([a(n) for n in range(1, 33)]) # Michael S. Branicky, Sep 27 2021

a(n) = |{k : A091333(k) = n}|.

A363968 Least number of 1's needed to represent n using only additions +, subtractions -, multiplications *, divisions /, concatenations # and parentheses ().

Original entry on oeis.org

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

Offset: 0

Bernard Schott, Jun 30 2023

Fractions are not allowed as intermediate results.
The unique difference with A362471 is that concatenation is here allowed; in fact, in A362471, concatenation is only allowed for getting repunits as 111 = 1#1#1 but not for getting other integers.
Also, for example, the concatenation of 5 and -3 is not possible, so it should not be interpreted as 5-3 = 2.
The first differences with A362471 in the data appear at n = 16, 19, 20, 21, 29, ... see Example section.

			For n = 16, 16 = 1 # ((1+1)*(1+1+1)), so a(16) = 6 while A362471(16) = 7.
For n = 19, 19 = 1 # (11-1-1), so a(19) = 5 while A362471(19) = 6.
For n = 20, 20 = (1+1) # (1-1), so a(20) = 4 while A362471(20) = 5.
For n = 31, 31 = (1+1+1) # (1), so a(31) = 4 while A362471(31) = 7.
For n = 43, 43 = (1+1)*((1+1) # (1)) + 1, so a(43) = 6 while A362471(43) = 7.

Michael S. Branicky, Table of n, a(n) for n = 0..10000.
Index to sequences related to the complexity of n.

Cf. A002275, A005245, A091333, A133344, A362471, A362626.

|a(n+1) - a(n)| <= 1; improved by Pontus von Brömssen, Jun 30 2023
a(n) <= A362471(n).
a(n) <= Sum_{k=1..m} a(dk), where d1d2..dm are the decimal digits of n. - Michael S. Branicky, Jun 30 2023

a(72) and beyond from Michael S. Branicky, Jun 30 2023

A373446 Number of distinct ways of expressing n using only addition, multiplication (with all factors greater than 1), necessary parentheses, and the number 1.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 6, 7, 10, 10, 18, 19, 27, 30, 50, 53, 80, 85, 133, 146, 209, 223, 350, 382, 544, 597, 886, 962, 1385, 1507, 2197, 2426, 3422, 3740, 5413, 5941, 8295, 9159, 12994, 14298, 19947, 21982, 30763, 34111, 47005, 51895, 72202, 79974, 109468, 121545, 167032, 185276, 252534, 280427, 382274, 425703, 575650, 640243, 867942

Offset: 1

Daniel W. Grace, Jun 05 2024

Expressions that are the same after commuting their terms are not considered distinct from one another.

Parentheses are used around a sum which is being multiplied, but not otherwise.

			a(10)=10, as 10 can be expressed in the following ways:
  1+1+1+1+1+1+1+1+1+1
  (1+1)*(1+1)+1+1+1+1+1+1
  (1+1)*(1+1)+(1+1)*(1+1)+1+1
  (1+1)*(1+1)*(1+1)+1+1
  (1+1)*(1+1+1)+1+1+1+1
  (1+1)*(1+1+1)+(1+1)*(1+1)
  (1+1)*(1+1+1+1)+1+1
  (1+1+1)*(1+1+1)+1
  (1+1)*(1+1+1+1+1)
  (1+1)*((1+1)*(1+1)+1).

Pontus von Brömssen, Table of n, a(n) for n = 1..10000

Cf. A002095, A005245, A214833.

from itertools import count,islice
from collections import Counter
from math import comb
from sympy import divisors
def euler_transform(x):
    xlist = []
    z = []
    y = []
    for n,x in enumerate(x,1):
        xlist.append(x)
        z.append(sum(d*xlist[d-1] for d in divisors(n)))
        yy = (z[-1]+sum(zz*yy for zz,yy in zip(z,reversed(y))))//n
        yield yy
        y.append(yy)
def factorizations(n,fmin=2):
    if n == 1:
        yield []
        return
    for d in divisors(n,generator=True):
        if d < fmin: continue
        for f in factorizations(n//d,d):
            yield [d]+f
def A373446_generator():
    alist = []
    def bgen():
        blist = []
        for n in count(1):
            b = 0
            for p in factorizations(n):
                if len(p) == 1: continue
                m = 1
                for k,c in Counter(p).items():
                    m *= comb(alist[k-1]-blist[k-1]+c-1,c)
                b += m
            yield b
            blist.append(b)
    for a in euler_transform(bgen()):
        yield a
        alist.append(a)
print(list(islice(A373446_generator(),60))) # Pontus von Brömssen, Jun 13 2024

a(n) >= a(n-1) since, if "+1" is appended to each expression used to calculate a(n-1), then each of the resulting expressions equate to n and are distinct from each other. There may or may not be other ways to express n that do not include an isolated "+1", hence the greater-than possibility.

A104233 Positive integers which have a "compact" representation using fewer decimal digits than just writing the number normally.

Original entry on oeis.org

125, 128, 216, 243, 256, 343, 512, 625, 729, 1000, 1015, 1016, 1017, 1018, 1019, 1020, 1021, 1022, 1023, 1024, 1025, 1026, 1027, 1028, 1029, 1030, 1031, 1032, 1033, 1080, 1089, 1125, 1152, 1156, 1215, 1225, 1250, 1280, 1287, 1288, 1289, 1290, 1291, 1292, 1293, 1294

Offset: 1

Jack Brennen, Apr 01 2005

You are allowed to use the following symbols as well:
  ( ) grouping
  + addition
  - subtraction
  * multiplication
  / division
  ^ exponentiation
Note that 1015 to 1033 are all representable in the form 4^5-d or 4^5+d, where d is a single digit.
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, Apr 02 2005
From Bernard Schott, Feb 10 2021: (Start)
These numbers are called "entiers compressibles" in French.
There are no 1-digit or 2-digit terms.
The 3-digit terms are all of the form m^q, with 2 <= m, q <= 9.
The 4-digit terms are of the form m^q with m, q > 1, or of the form m^q+-d or m^q*r with m, q, r > 1, d >= 0, and m, q, r, d are all digits (see examples where [...] is a corresponding "compact" representation). (End)

			From _Bernard Schott_, Feb 10 2021: (Start)
a(1) = 125 = [5^3] = 5*5*5 is the smallest cube.
a(5) = 256 = [2^8] = [4^4] = 16*16 is the smallest square.
a(6) = 343 = [7^3] is the smallest palindrome.
a(15) = 1019 = [4^5-5] is the smallest prime.
6555 = [3^8-5] = [35^2] = T(49) = 49*50/2 is the smallest triangular number.
362880 = 9! = [70*72^2] = [8*(6^6-6^4)] is the smallest factorial.
The smallest zeroless pandigital number 123456789 = [(10^10-91)/81] = [3*(6415^2+38)] is a term. (End)
The largest pandigital number 9876543210 = [(8*10^11+10)/81] = [(8*10^11+10)/9^2] = [5*(15^5+67)*51^2] is also a term. - _Bernard Schott_, Apr 20 2022

R. K. Guy, Unsolved Problems Number Theory, Sect. F26.

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.
Diophante, A164, Les entiers compressibles (in French).
R. K. Guy, Some suspiciously simple sequences, Amer. Math. Monthly 93 (1986), 186-190; 94 (1987), 965; 96 (1989), 905.
Eric Weisstein's World of Mathematics, Integer Complexity
Index to sequences related to the complexity of n

Cf. A036057, A005245, A003313, A076142, A076091, A061373, A005421, A064097, A005520, A025280, A003037.

More terms from Bernard Schott, Feb 10 2021

Missing terms added by David A. Corneth, Feb 14 2021

A244744 Smallest number with additive excess n.

Original entry on oeis.org

46, 253, 649, 6049, 69989, 166213, 551137, 9064261, 68444596, 347562415, 612220081

Offset: 1

Juan Arias-de-Reyna, Jul 05 2014

Let m = Product p_j ^ a_j be the standard prime factorization of m. The additive excess of m is defined to be Sum||p_j ^ a_j|| - ||m|| where ||m||=A005245(n) is the complexity of n.

The n-th term of this sequence is the least number m with additive excess n.

			||253||=17, ||11||=8, ||23||=11.  11 + 8 - 17 = 2. The second term is 253.

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.

Cf. A244743, A005245.

A346742 Numbers that may be built from fewer ones using floor(j/k) in addition to +, -, and *.

Original entry on oeis.org

1860043, 3198487, 4782847, 5580129, 6111571, 9300217, 9566302, 9595461, 9595462, 9654511, 10678027, 12725059, 12843157, 13551745, 14349271, 14614627, 16740391, 17094685, 18334713, 18334714, 19220449, 27900651, 28698178, 28701094, 29494975, 31620739, 32034081, 33484063, 34100797, 35872267, 37998031

Offset: 1

Glen Whitney, Sep 28 2021

nonn

Consider an integer complexity measure c(n) which is the number of ones required to build n using +, -, *, and "floor division" which for convenience will be written in this entry (after Python notation) j//k = floor(j/k). In other words, c(n) is defined identically to A091333(n) except that this floor division is also allowed, and identically to the complexity b(n) described in A348069 except that division is extended to all pairs of natural numbers by taking the floor of the quotient. Clearly for all n, c(n) <= b(n) <= A091333(n). This sequence lists the integers k for which c(k) < A091333(k).
Because of the inequality c(n) <= b(n) <= A091333(n), every entry in A348069 will eventually appear in this sequence. For example, the first term of A348069 is 50221174 = (7*3^15)//2, so we have c(50221174) = 53, b(50221174) = 54, and A091333(50221174) = 55.
The extended domain of division means that terms of this sequence are much more frequent than A348069, but it's still quite rare for division to provide more compact expressions for natural numbers (except in the presence of exponentiation, see A348089).

			The smallest n for which c(n) as defined in the comments is strictly less than A091333(n) is 1860043, because 1860043 = (7*3^12)//2 which requires c(7) + 12*c(3) + c(2) = 6 + 12*3 + 2 = 44 ones to express with these operations, whereas A091333(1860043) = A005245(1860043) = 45 by virtue of the minimal expression 1860043 = 2(2^2*5*7(3^4(3^4+1)+1)+1)+1 requiring 2+2*2+5+6+3*4+3*4+1+1+1+1 = 45 ones. Hence, the first term in this sequence is 1860043.
The next three terms with their respective minimal expressions:
3198487 = (3^9(2^2*3^4+1))//2 [46 ones] = 2*3(3^2(2^2*3*5+1)(2^2*3^5-1)+2)+1 [47 ones] = 2*3(2(7(2^2*3+1)(2^2*3(3^5+1)+1)+1)+1)+1 [48 ones]. Thus n=319487 is the least n for which c(n) < A091333(n) < A005245(n).
4782847 = (3^5(2*3^9-1))//2 [47 ones] = 2*3(2*5(3^2(2^2*3^3(3^4+1)+1)+1)+1)+1 [48 ones]
5580129 = 3*1860043 = 3((7*3^12)//2) [47 ones] = 2^3(3*5*7(3^4(3^4+1)+1)+1)+1 [48 ones]. Note this example critically takes advantage of the fact that * and // are not associative.

Cf. A253177 and A348069.

Cf. A091333 and A005245 (other integer complexity measures).

A373701 Extension of Mahler-Popken complexity to the rationals. The minimal number of 1's required to build the n-th positive rational in the Cantor ordering using only +, /, and *.

Original entry on oeis.org

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

Offset: 1

Adil Soubki, Jun 13 2024

nonn

Since we do not require that rationals with denominator 1 be written in the form p/q (i.e., we allow them to be written as p), this reduces to A005245 in the case where q = 1.

			|    | rational   |  minimal expression         |    a(n) |
|---:|:-----------|:----------------------------|--------:|
|  1 | 1/1        | 1                           |       1 |
|  2 | 1/2        | 1/(1+1)                     |       3 |
|  3 | 2/1        | 1+1                         |       2 |
|  4 | 1/3        | 1/(1+1+1)                   |       4 |
|  5 | 3/1        | 1+1+1                       |       3 |
|  6 | 1/4        | 1/(1+1+1+1)                 |       5 |
|  7 | 2/3        | (1+1)/(1+1+1)               |       5 |
|  8 | 3/2        | 1+(1/(1+1))                 |       4 |
|  9 | 4/1        | 1+1+1+1                     |       4 |
| 10 | 1/5        | 1/(1+1+1+1+1)               |       6 |
| 11 | 5/1        | 1+1+1+1+1                   |       5 |
| 12 | 1/6        | 1/((1+1)*(1+1+1))           |       6 |
| 13 | 2/5        | (1+1)/(1+1+1+1+1)           |       7 |
| 14 | 3/4        | (1+1+1)/(1+1+1+1)           |       7 |
| 15 | 4/3        | 1+(1/(1+1+1))               |       5 |
| 16 | 5/2        | 1+1+(1/(1+1))               |       5 |
| 17 | 6/1        | (1+1)*(1+1+1)               |       5 |
| 18 | 1/7        | 1/((1+1+1)*(1+1) +1)        |       7 |
| 19 | 3/5        | (1+1+1)/(1+1+1+1+1)         |       8 |
| 20 | 5/3        | 1+((1+1)/(1+1+1))           |       6 |
| 21 | 7/1        | (1+1)*(1+1+1)+1             |       6 |
| 22 | 1/8        | 1/((1+1)*(1+1)*(1+1))       |       7 |
| 23 | 2/7        | (1+1)/((1+1)*(1+1+1)+1)     |       8 |
| 24 | 4/5        | (1+1+1+1)/(1+1+1+1+1)       |       9 |
| 25 | 5/4        | 1+(1/(1+1+1+1))             |       6 |
| 26 | 7/2        | 1+1+1+(1/(1+1))             |       6 |
| 27 | 8/1        | (1+1)*(1+1)*(1+1)           |       6 |
| 28 | 1/9        | 1/((1+1+1)*(1+1+1))         |       7 |
| 29 | 3/7        | (1+1+1)/((1+1+1)*(1+1) +1)  |       9 |
| 30 | 7/3        | 1+1+(1/(1+1+1))             |       6 |
| 31 | 9/1        | (1+1+1)*(1+1+1)             |       6 |
| 32 | 1/10       | 1/((1+1+1)*(1+1+1)+1)       |       8 |
| 33 | 2/9        | (1+1)/((1+1+1)*(1+1+1))     |       8 |
| 34 | 3/8        | (1+1+1)/((1+1)*(1+1)*(1+1)) |       9 |
| 35 | 4/7        | (1+1+1+1)/((1+1)*(1+1+1)+1) |      10 |
| 36 | 5/6        | (1/(1+1))+(1/(1+1+1))       |       7 |
| 37 | 6/5        | 1+(1/(1+1+1+1+1))           |       7 |

Dimitri Zucker, The Most Underrated Concept in Number Theory, Combo Class Youtube video.

Cf. A005245 (Mahler-Popken complexity).

Ordering used: A020652 (Cantor numerators), A020653 (Cantor denominators).

A382685 a(n) is the least integer k requiring any combination of at least n 1's or 2's to build using + and *.

Original entry on oeis.org

1, 3, 5, 7, 11, 19, 23, 43, 59, 107, 173, 283, 383, 719, 1103, 1439, 3019, 4283, 8563, 14207, 20719, 31667, 52919, 105838, 165749, 290219, 495359, 880799, 1529279, 2417399, 4085639, 6973259

Offset: 1

Zhining Yang, Jun 02 2025

Of the first 30 terms, all except a(1) and a(24) are primes.

			a(6) = 19 because 19 = 1 + (2 + 2 * 2 * 2 * 2 ), and 19 cannot be built with five 1 and 2's.
a(10) = 107 because 107 = 1 + (2 + 2 * 2 * (2 + 2 * 2 * (2 + 2 * 2 ))), and 107 cannot be built with nine 1 and 2's.

Shouen Wang, Using 1, 2, +, x, () to build positive integers, Chinese BBS.
Index to sequences related to the complexity of n

Cf. A005245, A005520.

N:= 10^6: # for terms <= N
M[1]:= {1,2}: T[1]:= M[1]: A:= 1:
for n from 2 do
  M[n]:= `union`(seq({seq(seq(x+y, x = select(`<=`,M[i],N-y)),y=M[n-i])},i=1..n/2),
                 seq({seq(seq(x*y, x = select(`<=`,M[i],N/y)),y=M[n-i])},i=1..n/2)) minus T[n-1];
  T[n]:= T[n-1] union M[n];
  if M[n] = {} then break fi;
  A:= A, min(M[n]);
od:
A; # Robert Israel, Jun 09 2025

num=1500;
b=Array[99999&,num];a={};
b[[1]]=b[[2]]=1;
r=1;
Do[Do[s=b[[k]]+b[[n/k]];If[s

a(28)-a(32) from Hongyang Cao, Jun 10 2025

cp's OEIS Frontend

A306560 Smallest number of 1's required to build n using +, *, ^ and tetration.

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A319975 Smallest number of complexity n with respect to the operations {1, shift, multiply}.

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A348083 Number of positive numbers that can be built with n ones using +, -, and *, and require at least n ones.

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A363968 Least number of 1's needed to represent n using only additions +, subtractions -, multiplications *, divisions /, concatenations # and parentheses ().

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A373446 Number of distinct ways of expressing n using only addition, multiplication (with all factors greater than 1), necessary parentheses, and the number 1.

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A104233 Positive integers which have a "compact" representation using fewer decimal digits than just writing the number normally.

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A244744 Smallest number with additive excess n.

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A346742 Numbers that may be built from fewer ones using floor(j/k) in addition to +, -, and *.

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A373701 Extension of Mahler-Popken complexity to the rationals. The minimal number of 1's required to build the n-th positive rational in the Cantor ordering using only +, /, and *.

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