

A168650


Integers that can be generated with a C/C++ expression that is shorter than their decimal representation.


3



1000, 2000, 3000, 4000, 5000, 6000, 7000, 8000, 9000, 10000, 11000, 12000, 13000, 14000, 15000, 16000, 17000, 18000, 19000, 20000, 21000, 22000, 23000, 24000, 25000, 26000, 27000, 28000, 29000, 30000, 31000, 32000, 33000, 34000, 35000
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OFFSET

1,1


COMMENTS

From Dmitry Kamenetsky, Jul 24 2015: (Start)
By "expression" we mean a string representing a piece of code in C/C++ that evaluates to a positive integer, where we assume for simplicity that the result is converted to an integer using the floor operation. An expression can use the binary operators available in those languages and the digits '0' to '9', and we also allow "AeB" for A*10^B.
For example: "A+B" evaluates to A plus B, "A*B" evaluates to A multiplied by B, "A/B" is floor(A/B), "A<<B" is A*2^B.
This sequence lists every integer n having an expression whose length is strictly less than the decimal length of n; of course every integer n has an expression whose length is the same as the decimal length of n, namely "n". In some sense the numbers in this sequence can be considered simple, since they have a low Kolmogorov complexity.
We assume that there are no rounding errors or integer overflow during the evaluation of the expression.(End)


LINKS

Dmitry Kamenetsky, Table of n, a(n) for n = 1..14238
Dmitry Kamenetsky, Plot showing the length of the shortest C/C++ expression that generates integers from 1 to 2000000.
Ed Pegg, Jr., Integer Complexity, 2004.
L. Staiger, The Kolmogorov complexity of real numbers, Theoretical Computer Science, pp. 455466, 2002.
E. Weisstein, Integer Complexity
Wikipedia, Kolmogorov Complexity
Index to sequences related to the complexity of n


EXAMPLE

1000 has 4 digits, but it can be generated with a 3digit expression "1e3". The integers 43000, 116666, 114688, 199997 are also in the sequence, since they can be generated using the expressions "43e3", "7e5/6", "7<<14", "2e53" respectively.


CROSSREFS

Cf. A005245, A117607, A118121, A168651, A168652.
Sequence in context: A004267 A319441 A202456 * A043491 A100988 A195163
Adjacent sequences: A168647 A168648 A168649 * A168651 A168652 A168653


KEYWORD

nonn,base


AUTHOR

Dmitry Kamenetsky, Dec 01 2009


EXTENSIONS

Name clarified by Dmitry Kamenetsky, Jul 24 2015


STATUS

approved



