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%I #39 Mar 20 2024 11:11:34
%S 2,2399,1801152661459,73885357344138503765443
%N Primes of the form prime(k)^k - PrimePi(k).
%C Searched up to k = 1000.
%C No additional entries up to k = 5000. - _Ethan Beihl_, Oct 15 2016
%C No additional entries up to k = 15000. - _Tyler Busby_, Mar 12 2024
%H Farideh Firoozbakht, <a href="https://primes.utm.edu/curios/page.php/14.html">Prime Curios: Prime(14)^14 - pi(14) is prime</a>.
%e 2 is in the sequence because 2 is prime and 2 = prime(1)^1 - PrimePi(1) = 2^1 - 0.
%e 2399 is in the sequence because 2399 is prime and 2399 = prime(4)^4 - PrimePi(4) = 7^4 - 2.
%e 1801152661459 is in the sequence because 1801152661459 is prime and 1801152661459 = prime(9)^9 - PrimePi(9) = 23^9 - 4.
%e 73885357344138503765443 is in the sequence because 73885357344138503765443 is prime and 73885357344138503765443 = prime(14)^14 - PrimePi(14) = 43^14 - 6.
%t Select[Map[Prime[#]^# - PrimePi@ # &, Range@ 1500], PrimeQ] (* _Michael De Vlieger_, Sep 26 2016 *)
%o (SageMath)
%o max_n = 20
%o seq = []
%o for n in range(1, max_n+1):
%o p = nth_prime(n)^n - prime_pi(n)
%o if is_prime(p):
%o seq.append(p)
%o print(seq)
%Y Cf. A000720, A062457.
%K nonn,more
%O 1,1
%A _Robert C. Lyons_, Sep 09 2016
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