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

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

Table of n, a(n) for n=1..46.

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

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

Crossrefs

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

Sequence in context: A028939 A195420 A080538 * A046759 A115938 A126895

Adjacent sequences: A104230 A104231 A104232 * A104234 A104235 A104236

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