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A005383
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Numbers n such that both n and (n+1)/2 are primes.
(Formerly M2492)
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79
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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
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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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
B. Cloitre, On the fractal behavior of primes, 2011 [Broken link]
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FORMULA
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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
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EXAMPLE
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Both 3 and (3+1)/2 = 2 are primes, both 5 and (5+1)/2 = 3 are primes. - Zak Seidov, Nov 19 2012
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MAPLE
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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
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MATHEMATICA
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Select[Prime[Range[1000]], PrimeQ[(# + 1)/2] &] (* Zak Seidov, Nov 19 2012 *)
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PROG
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(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
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CROSSREFS
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Cf. A005382, A057326, A057327, A057328, A057329, A057330, A005603.
A subsequence of A000040 which has A036570 as subsequence.
Cf. A010051, A065091, A048161, A036570.
Sequence in context: A324783 A032009 A032027 * A306826 A175257 A190423
Adjacent sequences: A005380 A005381 A005382 * A005384 A005385 A005386
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from David Wasserman, Jan 18 2002
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STATUS
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approved
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