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
Examples
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
References
- R. K. Guy, Unsolved Problems Number Theory, Sect. F26.
Links
- 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
Crossrefs
Extensions
More terms from Bernard Schott, Feb 10 2021
Missing terms added by David A. Corneth, Feb 14 2021
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