08/11/2014, 12:17 PM

Some variants :

In general f(n) = f(n-1) + f(n/a1) + f(n/a2) + ...

or f(n) = f(n-1) + f(n/a1) - f(n/a2) + f(n/a3) - ...

where 2<a1<a2<a3<... and all devisions are rounded to the smallest integer (floor). Also f(0) = 0 and f(1) = 1.

In particular

f(n) = f(n-1) + f(n/3) + f(n/5) + f(n/7) + f(n/9) + ...

and

g(n) = g(n-1) + g(n/3) - g(n/5) + g(n/7) - g(n/9) + ...

The growth rates of these 2 are fascinating.

Clearly this has number theoretic value.

also the sum 1/a1 + 1/a2 + ... is fascinating when it converges.

For instance

A(n) = A(n-1) + A(n/3)

B(n) = B(n-1) + B(n/4) + B(n/12)

Notice 1/3 = 1/4 + 1/12.

Does this imply that A(n) is close to B(n) ??

Many ideas and conjectures pop up.

regards

tommy1729

In general f(n) = f(n-1) + f(n/a1) + f(n/a2) + ...

or f(n) = f(n-1) + f(n/a1) - f(n/a2) + f(n/a3) - ...

where 2<a1<a2<a3<... and all devisions are rounded to the smallest integer (floor). Also f(0) = 0 and f(1) = 1.

In particular

f(n) = f(n-1) + f(n/3) + f(n/5) + f(n/7) + f(n/9) + ...

and

g(n) = g(n-1) + g(n/3) - g(n/5) + g(n/7) - g(n/9) + ...

The growth rates of these 2 are fascinating.

Clearly this has number theoretic value.

also the sum 1/a1 + 1/a2 + ... is fascinating when it converges.

For instance

A(n) = A(n-1) + A(n/3)

B(n) = B(n-1) + B(n/4) + B(n/12)

Notice 1/3 = 1/4 + 1/12.

Does this imply that A(n) is close to B(n) ??

Many ideas and conjectures pop up.

regards

tommy1729