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A261810 n and (2*n^2 + 2*n - 1) are primes.

%I A261810 #26 Mar 04 2025 07:36:41
%S A261810 2,3,5,11,23,59,71,113,131,137,149,179,227,257,263,269,293,317,347,
%T A261810 353,401,419,443,449,467,557,653,659,677,683,743,773,809,839,857,881,
%U A261810 911,929,947,977,1019,1049,1277,1301,1319,1433,1571,1697,1847,1871,1901,1913
%N A261810 n and (2*n^2 + 2*n - 1) are primes.
%C A261810 Primes p such that (number of divisors of p * sum of divisors of p * product of divisors of p - 1) is also a prime.
%C A261810 Primes p such that (A000005(p) * A000203(p) * A007955(p) - 1) is also a prime.
%C A261810 See similar sequences of type primes p such that x is also a prime for some x wherein tau(p) = A000005(p) = number of divisors of p, sigma(p) = A000203(p) = sum of divisors of p and pod(p) = A007955(p) = product of divisors of p:
%C A261810 A001359 (for x = tau(p) + sigma(p) - 1 and x = tau(p) + pod(p)),
%C A261810 A005382 (for x = tau(p) * pod(p) - 1),
%C A261810 A005384 (for x = sigma(p) + pod(p), x = tau(p) * sigma(p) - 1 and x = tau(p) * pod(p) + 1),
%C A261810 A023200 (for x = tau(p) + sigma(p) + 1),
%C A261810 A023204 (for x = tau(p) + sigma(p) + pod(p) and x = tau(p) * sigma(p) + 1),
%C A261810 A053182 (for x = sigma(p) * pod(p) + 1),
%C A261810 A053184 (for x = sigma(p) * pod(p) - 1),
%C A261810 A158526 (for x = tau(p) * sigma(p) * pod(p) + 1).
%C A261810 For n >= 3, a(n) == 5 mod 6. - _Robert Israel_, Sep 02 2015
%H A261810 Robert Israel, <a href="/A261810/b261810.txt">Table of n, a(n) for n = 1..10000</a>
%e A261810 3 and 2*3^2 + 2*3 - 1 = 23 are primes.
%p A261810 select(t -> isprime(t) and isprime(2*t^2 + 2*t-1), [2,3,seq(6*i-1,i=1..1000)]); # _Robert Israel_, Sep 02 2015
%t A261810 Select[Prime[Range[300]], PrimeQ[2 #^2 + 2 # - 1] &] (* _Vincenzo Librandi_, Sep 02 2015 *)
%o A261810 (Magma) [n: n in[1..10000] | IsPrime(n) and IsPrime(2*n*n + 2*n - 1)];
%o A261810 (PARI) is(n)=isprime(n)&&isprime(2*n^2 + 2*n - 1) \\ _Anders Hellström_, Sep 01 2015
%Y A261810 Cf. A000005, A000203, A007955.
%Y A261810 Cf. A001359, A005382, A005384, A023200, A023204, A053182, A053184, A158526.
%K A261810 nonn
%O A261810 1,1
%A A261810 _Jaroslav Krizek_, Sep 01 2015