Re: A306044(n)

[Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>]

On Wed, Nov 16, 2022 at 04:21:10PM +0100, Ruud H.G. van Tol wrote:
> 
> I added PARI-code to https://oeis.org/A306044
> "Powers of 2, 3 and 5."
> but I'm not happy with it.
> 
> {
> a(n)=
> my(f= [2, 3, 5]);  /* ordered co-primes (?) */
> for(i= 1, #f
> , my(p= (n-1)\f[i], d= -1, j= 0, m= 0);
>   while( j < n
>   , d++;
>     m= f[i] ^ (p+d);
>     j= 1; for(k=1, #f, j+= logint(m, f[k])); /* j= index */
>     if( (j > n) && 0 == d, j=0; p--; d=-1) /* retry */
>   );
>   if(j==n, return(m))  /* found */
> )
> }
> 
> The initial value of 'p' is a (simple) guess.

Seems to me you want to invert

b(n)=logint(n,2)+logint(n,3)+logint(n,5)+1;

an approximation is thus

h(n)=exp((n-1)/(1/log(2)+1/log(3)+1/log(5)))

This function is a good approximation of a(n):

g(n)=my(f=[2,3,5],e,a=round(exp((n+1)/(1/log(2)+1/log(3)+1/log(5))),&e));vecmax([p^logint(a,p)|p<-f]);

Cheers
Bill