cp's OEIS Frontend

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

%I A104233 #64 May 04 2025 04:56:48
%S A104233 125,128,216,243,256,343,512,625,729,1000,1015,1016,1017,1018,1019,
%T A104233 1020,1021,1022,1023,1024,1025,1026,1027,1028,1029,1030,1031,1032,
%U A104233 1033,1080,1089,1125,1152,1156,1215,1225,1250,1280,1287,1288,1289,1290,1291,1292,1293,1294
%N A104233 Positive integers which have a "compact" representation using fewer decimal digits than just writing the number normally.
%C A104233 You are allowed to use the following symbols as well:
%C A104233   ( ) grouping
%C A104233   + addition
%C A104233   - subtraction
%C A104233   * multiplication
%C A104233   / division
%C A104233   ^ exponentiation
%C A104233 Note that 1015 to 1033 are all representable in the form 4^5-d or 4^5+d, where d is a single digit.
%C A104233 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
%C A104233 From _Bernard Schott_, Feb 10 2021: (Start)
%C A104233 These numbers are called "entiers compressibles" in French.
%C A104233 There are no 1-digit or 2-digit terms.
%C A104233 The 3-digit terms are all of the form m^q, with 2 <= m, q <= 9.
%C A104233 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)
%D A104233 R. K. Guy, Unsolved Problems Number Theory, Sect. F26.
%H A104233 J. Arias de Reyna, <a href="/A005245/a005245_1.pdf">Complejidad de los números naturales</a>, Gaceta R. Soc. Mat. Esp., 3 (2000), 230-250. [Cached copy, with permission]
%H A104233 J. Arias de Reyna, <a href="https://arxiv.org/abs/2111.03345">Complexity of natural numbers</a>, arXiv:2111.03345 [math.NT], 2021.
%H A104233 Diophante, <a href="http://www.diophante.fr/problemes-par-themes/arithmetique-et-algebre/a1-pot-pourri/1861-a164-les-entiers-compressibles">A164, Les entiers compressibles</a> (in French).
%H A104233 R. K. Guy, <a href="http://www.jstor.org/stable/2323338">Some suspiciously simple sequences</a>, Amer. Math. Monthly 93 (1986), 186-190; 94 (1987), 965; 96 (1989), 905.
%H A104233 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IntegerComplexity.html">Integer Complexity</a>
%H A104233 <a href="/index/Com#complexity">Index to sequences related to the complexity of n</a>
%e A104233 From _Bernard Schott_, Feb 10 2021: (Start)
%e A104233 a(1) = 125 = [5^3] = 5*5*5 is the smallest cube.
%e A104233 a(5) = 256 = [2^8] = [4^4] = 16*16 is the smallest square.
%e A104233 a(6) = 343 = [7^3] is the smallest palindrome.
%e A104233 a(15) = 1019 = [4^5-5] is the smallest prime.
%e A104233 6555 = [3^8-5] = [35^2] = T(49) = 49*50/2 is the smallest triangular number.
%e A104233 362880 = 9! = [70*72^2] = [8*(6^6-6^4)] is the smallest factorial.
%e A104233 The smallest zeroless pandigital number 123456789 = [(10^10-91)/81] = [3*(6415^2+38)] is a term. (End)
%e A104233 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
%Y A104233 Cf. A036057, A005245, A003313, A076142, A076091, A061373, A005421, A064097, A005520, A025280, A003037.
%K A104233 nonn,base
%O A104233 1,1
%A A104233 _Jack Brennen_, Apr 01 2005
%E A104233 More terms from _Bernard Schott_, Feb 10 2021
%E A104233 Missing terms added by _David A. Corneth_, Feb 14 2021