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A176045 Numbers n such that n-1 and 2*n-1 are both prime.

%I A176045 #18 Sep 08 2022 08:45:52
%S A176045 3,4,6,12,24,30,42,54,84,90,114,132,174,180,192,234,240,252,282,294,
%T A176045 360,420,432,444,492,510,594,642,654,660,684,720,744,762,810,912,954,
%U A176045 1014,1020,1032,1050,1104,1224,1230,1290,1410,1440,1452,1482,1500,1512,1560
%N A176045 Numbers n such that n-1 and 2*n-1 are both prime.
%C A176045 Also numbers n such that all eigenvalues of the n X n matrix M_n defined in A176043 are prime. The eigenvalues are 2*n-1, and n-1 with multiplicity n-1.
%C A176045 a(n)^2 = p^2 + q, where both p and q are primes. These are the only squares of this form, and which always yields q > p with a(n) - 1 = p = A005384(n) and 2*a(n) - 1 = q = A005385(n), for the same n. Also: a(n) = q - p; p + q + a(n) = 2q = A194593(n+1); and p*q = A156592  - _Richard R. Forberg_, Mar 04 2015
%H A176045 Harvey P. Dale, <a href="/A176045/b176045.txt">Table of n, a(n) for n = 1..1000</a>
%F A176045 a(n) = A005384(n)+1.
%F A176045 a(n) = 2*A124485(n-1) for n > 1.
%e A176045 6-1 = 5 and 2*6-1 = 11 are both prime, so 6 is in the sequence. 7-1 = 6 and 2*7-1 = 13 are not both prime, so 7 is not in the sequence.
%e A176045 p = 3, q = 7; p^2 + q = 16, a(n) = sqrt(16) = 4. - _Richard R. Forberg_, Mar 04 2015
%p A176045 with(numtheory):for n from 2 to 2000 do:if type((2*n-1),prime)=true and type((n-1),prime)=true then print(n):else fi:od:
%t A176045 Select[Prime[Range[250]],PrimeQ[2#+1]&]+1 (* _Harvey P. Dale_, Jul 31 2013 *)
%o A176045 (Magma) [ n: n in [2..1600] | IsPrime(n-1) and IsPrime(2*n-1) ]; // _Klaus Brockhaus_, Apr 19 2010
%o A176045 (PARI) isok(n) = isprime(n-1) && isprime(2*n-1); \\ _Michel Marcus_, Apr 06 2016
%Y A176045 Cf. A176043, A005384 (Sophie Germain primes), A005385 (Safe Primes),  A124485 (2*n-1 and 4*n-1 are prime).
%K A176045 nonn
%O A176045 1,1
%A A176045 _Michel Lagneau_, Apr 07 2010
%E A176045 Edited and 1482 inserted by _Klaus Brockhaus_, Apr 19 2010