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The odd harmonics of a distorted (non-sinusoidal) periodic signal are harmonics whose frequency is an odd integer multiple of the fundamental frequency of the distorted signal (which is the same as the frequency of the fundamental component). So, their order is given by:
Another refinement is to initially list odd numbers only, (3, 5, ..., n), and count in increments of 2p in step 3, thus marking only odd multiples of p. This actually ...
Every second number in these triplets will be a multiple of 3, because numbers of the form 3 + 6k are all odd multiples of 3. Thus all the numbers coprime with the first two primes (2 and 3) will be generated by repeated additions of 6, starting from {1, 5}: 1, 5 ; 7, 11 ; 13, 17 ; ...
3 is the first Mersenne prime, as well as the second Mersenne prime exponent and the second double Mersenne prime exponent, for 7 and 127, respectively. 3 is also the first of five known Fermat primes, which include 5, 17, 257, and 65537.
The sequence of double factorials for odd n = 1, 3, 5, 7, 9, ... the following definition of the integer-valued multiple factorial functions (multifactorials), or ...
3 · P 8 4 800 Hz: 0.0 ¢ Play ⓘ 9 th: Pythagorean major second harmonic ninth 3 · P 8 + M 2 5 400 Hz: 203.9 ¢ Play ⓘ 10 th: just major third: 3 · P 8 + M 3 6 000 Hz: 386.3 ¢ Play ⓘ 11 th: lesser undecimal tritone, undecimal semi-augmented fourth: 3 · P 8 + A 4: 6 600 Hz: 551.3 ¢ Play ⓘ 12 th: perfect fifth: 3 · P 8 + P 5 7 200 ...
Squares of odd numbers are odd, and are congruent to 1 modulo 8, since (2n + 1) 2 = 4n(n + 1) + 1, and n(n + 1) is always even. In other words, all odd square numbers have a remainder of 1 when divided by 8. Every odd perfect square is a centered octagonal number. The difference between any two odd perfect squares is a multiple of 8.
The Liouville function λ(n) is 1 if Ω(n) is even, and is -1 if Ω(n) is odd. The Möbius function μ(n) is 0 if n is not square-free. Otherwise μ(n) is 1 if Ω(n) is even, and is −1 if Ω(n) is odd. A sphenic number has Ω(n) = 3 and is square-free (so it is the product of 3 distinct