login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A005383 Numbers n such that both n and (n+1)/2 are primes.
(Formerly M2492)
67
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 (list; graph; refs; listen; history; text; internal format)
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

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

B. Cloitre, On the fractal behavior of primes, 2011 [Broken link]

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

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) select(n->isprime(n\2+1), primes(100)[2..100]) \\ 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

l=[]

for n in range(2, 5000):

....if isprime(n) and isprime((n + 1)/2): l+=[n, ]

print l # Indranil Ghosh, Mar 17 2017

CROSSREFS

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: A283844 A032009 A032027 * A175257 A190423 A278024

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

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified September 20 21:08 EDT 2017. Contains 292293 sequences.