A176045 - OEIS

%I #18 Sep 08 2022 08:45:52

%S 3,4,6,12,24,30,42,54,84,90,114,132,174,180,192,234,240,252,282,294,

%T 360,420,432,444,492,510,594,642,654,660,684,720,744,762,810,912,954,

%U 1014,1020,1032,1050,1104,1224,1230,1290,1410,1440,1452,1482,1500,1512,1560

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

%C 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 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 Harvey P. Dale, <a href="/A176045/b176045.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = A005384(n)+1.

%F a(n) = 2*A124485(n-1) for n > 1.

%e 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 p = 3, q = 7; p^2 + q = 16, a(n) = sqrt(16) = 4. - _Richard R. Forberg_, Mar 04 2015

%p 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 Select[Prime[Range[250]],PrimeQ[2#+1]&]+1 (* _Harvey P. Dale_, Jul 31 2013 *)

%o (Magma) [ n: n in [2..1600] | IsPrime(n-1) and IsPrime(2*n-1) ]; // _Klaus Brockhaus_, Apr 19 2010

%o (PARI) isok(n) = isprime(n-1) && isprime(2*n-1); \\ _Michel Marcus_, Apr 06 2016

%Y Cf. A176043, A005384 (Sophie Germain primes), A005385 (Safe Primes),  A124485 (2*n-1 and 4*n-1 are prime).

%K nonn

%O 1,1

%A _Michel Lagneau_, Apr 07 2010

%E Edited and 1482 inserted by _Klaus Brockhaus_, Apr 19 2010