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

1,1

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

Links

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

Formula

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

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

Example

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

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 *)

Prog

(Magma) [ n: n in [2..1600] | IsPrime(n-1) and IsPrime(2*n-1) ]; // Klaus Brockhaus, Apr 19 2010
(PARI) isok(n) = isprime(n-1) && isprime(2*n-1); \\ Michel Marcus, Apr 06 2016

Crossrefs

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

Sequence in context: A095016 A358358 A160684 * A350299 A255733 A137333

Adjacent sequences: A176042 A176043 A176044 * A176046 A176047 A176048

Keyword

nonn

Author

Michel Lagneau, Apr 07 2010

Extensions

Edited and 1482 inserted by Klaus Brockhaus, Apr 19 2010

Status

approved