[seqfan] structure & sequences defined by "exotic multiplication"

[Maximilian Hasler]

> the sequence may be more interesting if "o" were associative, is there a
> list of binary associative functions?

I had an idea for an "exotic" associative operation ;
it's based on a very simple idea, so I expect it to be studied
somewhere, but as far as I can recall, I never heard of it.

The idea is to consider the bijection of positive integers (or fractions)
with polynomials of [nonnegative] integer coefficients :

n = product( p_i ^ e_i )  <-->  P_n = sum( e_i * X^i )

(where we should probably number the primes starting from 0 on:
p_0 = 2, p_1 = 3, ...)

Then addition of these polynomials corresponds to multiplication of numbers
( P_{mn} = P_m + P_n )
while multiplication of polynomials (convolution product) is the new
operation @,
 P_m * P_n =: P_{ m @ n }

Or otherwise said, if m = product( p_i ^ f_i ),

m @ n = product( p_k ^ (sum_{i=0..k} e_{k-i} * f_i)


Then we have commutativity and associativity of @,
and distributivity of @ over * :
k @ (m * n) = ( k @ m ) * ( k @ n ).

The prime 2 (corresp. to polynomial X^0) is the neutral element for @,
and repeated composition of 3 will yield all primes :
3 @ 3 (=: 3 ^^ 2) = 5
5 @ 3 (= 3 ^^ 3) = 7
7 @ 3 (= 3 ^^ 4) = 11
etc.

I think it could be worth while studying the structure resulting from
this operation
(it turns the multiplicative group of fractions into a ring)
in particular in the context of multiplicative functions.

Of course this operation also leads to some new sequences. Obviously,
@-multiplication by a prime just corresponds to shifting the prime
factors to the n-th subsequent prime.

n @ 3 : (= A003961 : multiplicative with a(p(k)) = p(k+1))
1, 3, 5, 9, 7, 15, 11, 27, 25, 21, 13, 45, 17, 33, 35, 81, 19, 75, 23,
63, 55, 39, 29, 135, 49, 51, 125, 99, 31, 105, ...

n @ 4 : amounts to taking squares (exponents of prime factors are doubled)

n @ 5 : (= A045966 except for the initial term)
1, 5, 7, 25, 11, 35, 13, 125, 49, 55, 17, 175, 19, 65, 77, 625, 23,
245, 29, 275, 91, 85, 31, 875, 121, 95, 343, 325, 37, 385,...

n @ 6 : (not in OEIS)
1, 6, 15, 36, 35, 90, 77, 216, 225, 210, 143, 540, 221, 462, 525,
1296, 323, 1350, 437, 1260, 1155, 858, 667, 3240, 1225, 1326, 3375,
2772, 899, 3150, ...

less trivial : n ^^ 2 = n @ n : (not yet in OEIS).

1, 2, 5, 16, 11, 90, 17, 512, 625, 550, 23, 6480, 31, 1666, 2695,
65536, 41, 101250, 47, 110000, 10285, 5566, 59, 1866240, 14641, 10478,
1953125, 653072, 67, 1212750

etc... (n ^^ 3, ...)

But once again, I suspect this has already been studied.
Does anyone know about this, or have a reference ?

Regards,
Maximilian