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) = 2*A005382(n) - 1. - Zak Seidov, Nov 19 2012
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
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from David Wasserman, Jan 18 2002
Name changed by Jianing Song, Nov 27 2021
STATUS
approved