Between addition and product ( pic )

[tommy1729]

Here is a pic showing An operator / function between addition and product.

It maps N x N to N.

It is commutative and associative.

Regards

Tommy1729

[tommy1729]

(0,n) -> 0
(1,n) -> n

Since that is trivial , it is not written in the picture.
So the table starts at 2.

Regards

Tommy1729

[deepblue]

Its nice that they are all Natural numbers too

How are they being calculated?

the first 3 lines can be found in oeis databank



https://oeis.org/search?q=4%2C5%2C7%2C9%...&go=Search

https://oeis.org/search?q=5%2C8%2C9%2C13...&go=Search

https://oeis.org/search?q=7%2C9%2C12%2C1...&go=Search

[tommy1729]

My apologies for the delay.

I have been thinking about this rather than writing.

I would like to know more before i say more.

Its all very new.

But i can say this is about hyperoperators and a half iterate of 2^(z+1).

Regards

Tommy1729

[Gottfried]

The rowwise second order differences are not monotonously increasing. So what is this about?

[Catullus]

Six years later, and he has still not explained here how it works.

[Gottfried]

Six years later, and he has still not explained here how it works.

Perhaps interesting - an answer by me (2015) for half-addition at MSE

Gottfried

[tommy1729]

I had Many ideas that changed over time.

No proper solid definition or the sequence got stuck ( not definable or paradoxal ).


But one of the last versions of the idea was this ( doesnt match With the fitst picture But it was close. )

see pictures ;



Regards 

tommy1729

[tommy1729]

I noticed staying to close to Sum gives issues.
Holding on to product longer works better.

There are still Many questions and fear of asympt to product or sum rather than in between…

Also one wonders If this approximates bennet base sqrt 2 or so.
See pictures. 

Regards

Tommy1729

[tommy1729]

I noticed staying to close to Sum gives issues.
Holding on to product longer works better.

There are still Many questions and fear of asympt to product or sum rather than in between…

Also one wonders If this approximates bennet base sqrt 2 or so.
See pictures. 

Regards

Tommy1729

Notice the potential trend : f(x,x) = 5x - 6.

—-

Also one wonders how often values occur in the table.

If they never occur they are The analogue Numbers to primes.

And then one wonders About the growth of this sequence.

O ( 5 x / semilog(x) ) ??


Regards

Tommy1729