A089485 - OEIS

%I #19 Feb 13 2023 14:18:33

%S 3,5,15,35,55

%N Numbers k such that k^4 + 4^k = A001589(k) is a semiprime.

%C For n = 2*k + 1, n^4 + 4^n = (n^2 + n*2^(k + 1) + 2^n) * (n^2 - n*2^(k + 1) + 2^n) The sequence gives those values of n for which both parentheses are primes. No further terms were found for k<=5000.

%C a(6) > 120000, if it exists. - _Tyler Busby_, Feb 13 2023

%H Ignacio Larrosa Cañestro et al., <a href="https://groups.google.com/d/msg/sci.math/ukqOhcTAoyU/rmeXZwI6oXAJ">Find all primes of the form 4^n + n^4.</a> Discussion in newsgroup sci.math (2003).

%e a(1)=3 because 3^4+4^3=145=5*29, a(2)=5 because 5^4+4^5=1649=17*97.

%t Select[Range[60],PrimeOmega[#^4+4^#]==2&] (* _Harvey P. Dale_, Jul 31 2020 *)

%o (PARI) for(k=0,5000,my(n=2*k+1,p1=n^2+n*2^(k+1)+2^n,p2=n^2-n*2^(k+1)+2^n);if(ispseudoprime(p1)&&ispseudoprime(p2),print1(n,", "))) \\ _Hugo Pfoertner_, Jul 24 2019

%Y Cf. A001589.

%K nonn,more

%O 1,1

%A _Hugo Pfoertner_, Nov 11 2003