Riffs & Rotes

[Jon Awbrey]

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R&R.  Note 5

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Cf: http://www.research.att.com/projects/OEIS?Anum=A107532

Re: R&R 4.  http://stderr.org/pipermail/inquiry/2005-May/002737.html
In: R&R.    http://stderr.org/pipermail/inquiry/2005-May/thread.html#2714

Neil, SeqFans,

Having just noticed yet another potential ambiguity
in the term "j^th power part", let me try it thusly:

The right diagonal labeled by the prime power of the form j:k = (prime(j))^k
contains the j^th power primes in the factorization raised to the k^th power.
For example, the right diagonal labeled by the number 2 = 1:1 = (prime(1))^1
contains the power-free parts of each positive integer, specifically A055231,
and the right diagonal labeled by the number 4 = 1:2 = (prime(1))^2 contains
the squares of the square-free parts of positive integers.  In general, then,
the right diagonal labeled by m = (j_i : k_i)_i = Product_i prime(j_i)^(k_i)
contains the product over i of the (j_i)th power primes in the factorization
raised to the (k_i)th powers.

For example, the operator 5 = 3:1 extracts the 3rd power primes
in the factorization of each n and raises them to the 1st power,
thus sending 8 = 1:3 to 2 = 1:1, 27 = 2:3 to 3 = 2:1, and so on.

Jon Awbrey

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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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