Type of metrical foot can contribute a poetical nuance to certain numbers, as follows.

Definition 1. A metrical number is an integer n greater than unity having at least two distinct exponents in its prime factorization.

Theorem 1. No square-free number is a metrical number.

Proof: All the exponents are unity, so there do not exist two distinct exponents. QED.

Then, for a given metrical number, the number of primes in its prime factorization is the number of syllables in the foot (with the order of the primes matching the order of the syllables, of course), and the exponent on a given prime in its prime factorization corresponds to the stress that that syllable receives. So, for example, both 2×3^2 and 5×19^7 represent an iamb, but the latter has greater stress on the stressed syllable.

Definition 2. A PROPER metrical number is a metrical number that is the smallest metrical number of its type.

So, for example, 2×3^2 is a proper metrical number, being the smallest representation of an iamb.

The first few metrical numbers are:

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 150, 1470.

The first few PROPER metrical numbers are:

12, 18, 60, 150, 1470.

Neither of these sequences is yet recorded in the online list of integer sequences.

keywords: poems, Number Theory

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