A089485 Numbers k such that k^4 + 4^k = A001589(k) is a semiprime.

3, 5, 15, 35, 55

Offset: 1

Author

Hugo Pfoertner, Nov 11 2003

Keywords

• nonn
• more

Comments

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.

a(6) > 120000, if it exists. - Tyler Busby, Feb 13 2023

Examples

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

Links

• Ignacio Larrosa Cañestro et al., Find all primes of the form 4^n + n^4. Discussion in newsgroup sci.math (2003).

Crossrefs

Cf. A001589.

Programs

Mathematica

Select[Range[60],PrimeOmega[#^4+4^#]==2&] (* Harvey P. Dale, Jul 31 2020 *)

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