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The first: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728 (sequence A000578 in the OEIS). A perfect power has a common divisor m > 1 for all multiplicities (it is of the form a m for some a > 1 and m > 1).
Jul 8, 2005: A112886: Triangle-free positive integers. ... For example, 1, 8, 27, 64, 125, 216, 343, ... is an example of a sequence with offset −1. Sometimes there ...
Mscgen (short for MSC generator) is a software tool for drawing message sequence charts [1] from a simple to manage text-based source file. Rendered charts can be output in PNG, SVG and PostScript, with hyperlink information in ismap format.
"subtract if possible, otherwise add": a(0) = 0; for n > 0, a(n) = a(n − 1) − n if that number is positive and not already in the sequence, otherwise a(n) = a(n − 1) + n, whether or not that number is already in the sequence.
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Cycles of the unit digit of multiples of integers ending in 1, 3, 7 and 9 (upper row), and 2, 4, 6 and 8 (lower row) on a telephone keypad. Figure 1 is used for multiples of 1, 3, 7, and 9. Figure 2 is used for the multiples of 2, 4, 6, and 8. These patterns can be used to memorize the multiples of any number from 0 to 10, except 5.
Erdős (1962) showed that for every sum-free sequence there exists an unbounded sequence of numbers for which () = where is the golden ratio, and he exhibited a sum-free sequence for which, for all values of , () = (/), subsequently improved to () = (/) by Deshouillers, Erdős and Melfi in 1999 and to () = (/) by Luczak and Schoen in 2000, who ...
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