A005383 Primes p such that (p+1)/2 is prime. (Formerly M2492)

3, 5, 13, 37, 61, 73, 157, 193, 277, 313, 397, 421, 457, 541, 613, 661, 673, 733, 757, 877, 997, 1093, 1153, 1201, 1213, 1237, 1321, 1381, 1453, 1621, 1657, 1753, 1873, 1933, 1993, 2017, 2137, 2341, 2473, 2557, 2593, 2797, 2857, 2917, 3061, 3217, 3253

Offset

1,1

Comments

Also, n such that sigma(n)/2 is prime. - Joseph L. Pe, Dec 10 2001; confirmed by Vladeta Jovovic, Dec 12 2002

Primes that are followed by twice a prime, i.e., are followed by a semiprime. (For primes followed by two semiprimes, see A036570.) - Zak Seidov, Aug 03 2013, Dec 31 2015

If A005382(n) is in A168421 then a(n) is a twin prime with a Ramanujan prime, A104272(k) = a(n) - 2. - John W. Nicholson, Jan 07 2016

Starting with 13 all terms are congruent to 1 mod 12. - Zak Seidov, Feb 16 2017

Numbers n such that both n and n+12 are terms are 61, 661, 1201, 4261, 5101, 6121, 6361 (all congruent to 1 mod 60). - Zak Seidov, Mar 16 2017

Primes p for which there exists a prime q < p such that 2q == 1 (mod p). Proof: q = (p + 1)/2. - David James Sycamore, Nov 10 2018

Prime numbers n such that phi(sigma(2n)) = phi(2n), excluding n=3 and n=5; as well as phi(sigma(3n)) = phi(3n), excluding n=3 only. - Richard R. Forberg, Dec 22 2020

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

R. P. Boas & N. J. A. Sloane, Correspondence, 1974

Benoit Cloitre, On the fractal behavior of primes, 2011.

Formula

a(n) = A129521(n)/A005382(n). - Reinhard Zumkeller, Apr 19 2007

A000035(a(n))*A010051(a(n))*A010051((a(n)+1)/2) = 1. - Reinhard Zumkeller, Nov 06 2012

a(n) = 2*A005382(n) - 1. - Zak Seidov, Nov 19 2012

a(n) = A005382(n) + phi(A005382(n)) = A005382(n) + A000010(A005382(n)). - Torlach Rush, Mar 10 2014

Example

Both 3 and (3+1)/2 = 2 are primes, both 5 and (5+1)/2 = 3 are primes. - Zak Seidov, Nov 19 2012

Maple

for n to 300 do
  X := ithprime(n);
Y := ithprime(n+1);
Z := 1/2 mod Y;
  if isprime(Z) then print(Y);
end if:
end do:
# David James Sycamore, Nov 11 2018

Mathematica

Select[Prime[Range[1000]], PrimeQ[(# + 1)/2] &] (* Zak Seidov, Nov 19 2012 *)

Prog

(MATLAB) LIMIT = 8000 % Find all members of A005383 less than LIMIT A = primes(LIMIT); n = length(A); %n is number of primes less than LIMIT B = 2*A - 1; C = ones(n, 1)*A; %C is an n X n matrix, with C(i, j) = j-th prime D = B'*ones(1, n); %D is an n X n matrix, with D(i, j) = (i-th prime)*2 - 1 [i, j] = find(C == D); A(j)
(Magma) [n: n in [1..3300] | IsPrime(n) and IsPrime((n+1) div 2) ]; // Vincenzo Librandi, Sep 25 2012
(PARI) A005383_list(n) = select(m->isprime(m\2+1), primes(n)[2..n]) \\ Charles R Greathouse IV, Sep 25 2012
(Haskell)
a005383 n = a005383_list !! (n-1)
a005383_list = [p | p <- a065091_list, a010051 ((p + 1) `div` 2) == 1]
-- Reinhard Zumkeller, Nov 06 2012
(Python)
from sympy import isprime
[n for n in range(3, 5000) if isprime(n) and isprime((n + 1)/2)]
# Indranil Ghosh, Mar 17 2017
(Sage)
[n for n in prime_range(3, 1000) if is_prime((n + 1) // 2)]
# F. Chapoton, Dec 17 2019

Crossrefs

Cf. A005382, A057326, A057327, A057328, A057329, A057330, A005603.

A subsequence of A000040 which has A036570 as subsequence.

Cf. A005385, A010051, A065091, A048161, A036570.

Sequence in context: A032009 A032027 A360863 * A306826 A175257 A190423

Adjacent sequences: A005380 A005381 A005382 * A005384 A005385 A005386

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from David Wasserman, Jan 18 2002

Name changed by Jianing Song, Nov 27 2021

Status

approved