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Primes of the form 2*p^k-1, where p is prime and k > 1.
2

%I #10 Feb 21 2024 08:27:06

%S 7,17,31,53,97,127,241,337,577,1249,3361,3697,4373,4801,6961,8191,

%T 10657,13121,23761,25537,31249,32257,33613,37537,49297,59581,64081,

%U 65521,77617,79201,89041,126001,131071,138337,153457,159013,171697,193441

%N Primes of the form 2*p^k-1, where p is prime and k > 1.

%C Includes the Mersenne primes > 3 (A000668) and primes of the form 2p^2-1 (A092057) as subsequences. Its union with A005383 gives A178490.

%H Robert Israel, <a href="/A178491/b178491.txt">Table of n, a(n) for n = 1..10000</a>

%e a(1) = 7 = 2*2^2-1 and a(2) = 17 = 2*3^2-1 are also in A092057, and a(3) = 31 = 2*2^4-1 = A000668(3), but a(4) = 53 = 2*3^3-1 is in neither of these subsequences.

%p N:= 10^6: # for terms <= N

%p P:= select(isprime,[2,seq(i,i=3..floor(sqrt((N+1)/2)),2)]):

%p R:= NULL:

%p for p in P do

%p for k from 2 do

%p v:= 2*p^k-1;

%p if v > N then break fi;

%p if isprime(v) then R:= R,v fi;

%p od od:

%p sort([R]); # _Robert Israel_, Feb 20 2024

%t Select[Prime[Range[20000]],!PrimeQ[(#+1)/2]&&Length[FactorInteger[(#+1)/2]]==1&]

%o (PARI) is_A178491(n) = isprime(n) & ispower((n+1)/2,,&n) & isprime(n)

%Y Cf. A000668, A005383, A092057, A178490.

%K nonn

%O 1,1

%A _Farideh Firoozbakht_ and _M. F. Hasler_, Oct 09 2010