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A176045 Numbers n such that n-1 and 2*n-1 are both prime. 2
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 (list; graph; refs; listen; history; text; internal format)
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: A095765 A095016 A160684 * A255733 A137333 A006719

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

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Last modified November 14 04:56 EST 2019. Contains 329110 sequences. (Running on oeis4.)