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A104233
Positive integers which have a "compact" representation using fewer decimal digits than just writing the number normally.
0
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
1,1
COMMENTS
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)
REFERENCES
R. K. Guy, Unsolved Problems Number Theory, Sect. F26.
LINKS
J. Arias de Reyna, Complejidad de los nĂºmeros 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
EXAMPLE
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
KEYWORD
nonn,base
AUTHOR
Jack Brennen, Apr 01 2005
EXTENSIONS
More terms from Bernard Schott, Feb 10 2021
Missing terms added by David A. Corneth, Feb 14 2021
STATUS
approved