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

3, 4, 6, 12, 24, 30, 42, 54, 84, 90, 114, 132, 174, 180, 192, 234, 240, 252, 282, 294, 360, 420, 432, 444, 492, 510, 594, 642, 654, 660, 684, 720, 744, 762, 810, 912, 954, 1014, 1020, 1032, 1050, 1104, 1224, 1230, 1290, 1410, 1440, 1452, 1482, 1500, 1512, 1560

Offset: 1

Author

Michel Lagneau, Apr 07 2010

Keywords

• nonn

Comments

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.

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

Examples

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

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

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

Programs

Magma

[ n: n in [2..1600] | IsPrime(n-1) and IsPrime(2*n-1) ]; // Klaus Brockhaus, Apr 19 2010

Maple

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:

Mathematica

Select[Prime[Range[250]],PrimeQ[2#+1]&]+1 (* Harvey P. Dale, Jul 31 2013 *)

PARI

isok(n) = isprime(n-1) && isprime(2*n-1); \\ Michel Marcus, Apr 06 2016

Formula

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

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

Extensions

Edited and 1482 inserted by Klaus Brockhaus, Apr 19 2010