Formula for A033941

[Jamil Silva]

The formula for https://oeis.org/A033941 could be a(n) = Ceiling(d[A002808 (n)]/2) - 1 ?

[M F Hasler]

On Sat, May 2, 2026 at 5:50 PM Jamil Silva <jmai...@gmail.com> wrote:

The formula for https://oeis.org/A033941 could be a(n) = Ceiling(d[A002808 (n)]/2) - 1 ?

 

Yes (except that formulas shouldn't use Mathematica code...).

The number of "proper" factorisations of any number n is 

 f(n) = (numdiv(n) + issquare(n))/2 - 1

as written in the PARI code, because the divisors are used in pairs

except for the middle divisor sqrt(n) if n is a square,

and we subtract 1 for the trivial factorisation 1*n which we want to exclude.

So you can write it indeed as ceiling( numdiv(n)/2 ) - 1, or even shorter: 

f(n) = (numdiv(n)-1)\2 , with the "floor division" operator \ of PARI,

i.e.: A033941(n) = floor((numdiv(A002808(n)) - 1)/2).

-M.