A003037 -id:A003037 - cp's OEIS frontend

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

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

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

A347983 Smallest number requiring n 1's to build using +, -, *, and ^.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 11, 13, 21, 39, 41, 43, 115, 173, 276, 413, 823, 1389, 1654

Offset: 1

Glen Whitney, Sep 22 2021

Until n = 10 the terms are equal to A003037(n) where subtraction is not allowed; that is the same value of n at which A255641 and A005520, which also differ only in allowing subtraction, diverge.

The values given are all of the exact ones available from the program posted with A091334, which ignores intermediate results over 2^65, but which nevertheless is provably exact for small values of n up to complexity 19. Running the same program with a larger complexity limit leads to the uncertain (but highly likely correct) values for a(20) through a(26): 3306, 3307, 8871, 22261, 31661, 69467, 155051. (These values were stable for different intermediate-result cutoffs from 2^33 through 2^65, supporting their likely correctness.)

			a(7) = 11 because 2=1+1, 3=1+1+1, 4=1+1+1+1, 5=1+1+1+1+1, 6=(1+1)(1+1+1), 7=(1+1)(1+1+1)+1, 8=(1+1)^(1+1+1), 9=(1+1+1)^(1+1), and 10=(1+1+1)^(1+1)+1, all requiring fewer than seven ones, whereas a minimal way of expressing 11 is (1+1+1)^(1+1)+1+1 with seven ones. (Subtraction does not actually play a necessary role in a minimal expression until 15=(1+1)^(1+1+1+1)-1, and does not affect the value of a(n) until n = 10 because 23=(1+1+1)(1+1)^(1+1+1)-1 would otherwise be the smallest number requiring ten ones.)

Least inverse (or records) of A091334.

Cf. least inverses A003037, A005520, A255641 of other such "complexity" measures.

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

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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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A347983 Smallest number requiring n 1's to build using +, -, *, and ^.

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