[ March 1, 2022 ]
Sequence and patterns with n and φ(n) both palindromes
( phi(n) or φ(n) being the Euler totient function )
An exploration by Alexandru Petrescu (email)

Sequence with n and φ(n) both palindromes

φ(n) is the Euler totient function –
number of positive integers less than n, coprime with n.

Two numbers are said to be coprime if there are no
common factors other than 1.

I checked only numbers n having at least 2 digits,
because all one digit numbers have this property.

The data for n<10⁹ are in the first table.

And so on

The following Pari/gp program (by PDG) recreates
the solutions from the first table and more. It builds
on the assumation that Alexandru's conjecture is true
and that there are only n's with odd number of digits.

Nextpalodd(k)= \
  n=digits(k); le2=length(n)\2; \
  swl=digits(fromdigits(n[1..le2+1])+1); m=fromdigits(concat(swl,Vecrev(swl[1..le2]))); \
return(m);
{
k=100; cnt=0;
for(i=1,oo,
   Nextpalodd(k); k=m; ned=digits(eulerphi(m));
   if(Vecrev(ned)==ned, cnt+=1; print(cnt," ",m," ",eulerphi(m)) );
   if(i%100000==0, print1(k,Strchr(13)) );
);
}

The last line could be deleted if you're not interested in the progress.
It also saves some running time.
Within 5 minutes you get the 89th term of the sequence.