[seqfan] turtle product

Neil Fernandez primeness at borve.org
Tue Dec 10 17:41:12 CET 2013

Hi sefans,

Noting that the set of turtle transforms is equivalent to the set of
integer sequences, we can define the (non-commutative) turtle product of
two integer sequences a(n) and b(n) as the sequence of Gaussian integers

a(n)#b(n) := c(n) = sum from 1 to n of a(n)*i^(b(n)).

Then the sequence

c(n) := sum from k=1 to n of pi(k)*i^(k(k-1)/2)

plotted at <http://borve.org/plots/1000.jpg>

can be called the turtle product of the primes and the triangular
numbers, with the latter starting at 0.

Plot of (primes)#(Fibonaccis):


This gives rise to questions such as:

what sequences commute with the primes?


Neil Fernandez

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