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%I M0849 #102 Feb 05 2024 00:56:58
%S 2,3,7,19,31,37,79,97,139,157,199,211,229,271,307,331,337,367,379,439,
%T 499,547,577,601,607,619,661,691,727,811,829,877,937,967,997,1009,
%U 1069,1171,1237,1279,1297,1399,1429,1459,1531,1609,1627,1657,1759,1867,2011
%N Primes p such that 2p-1 is also prime.
%C Sequence gives values of p such Sum_{i=1..p} gcd(p,i) = A018804(p) is prime. - _Benoit Cloitre_, Jan 25 2002
%C Let q = 2n-1. For these n (and q), the sum of two cyclotomic polynomials can be written as a product of cyclotomic polynomials and as a cyclotomic polynomial in x^2: Phi(q,x) + Phi(2q,x) = 2 Phi(n,x) Phi(2n,x) = 2 Phi(n,x^2). - _T. D. Noe_, Nov 04 2003
%C Primes in A006254. - _Zak Seidov_, Mar 26 2013
%C If a(n) is in A168421 then A005383(n) is a twin prime with a Ramanujan prime, A005383(n) - 2. If this sequence has an infinite number of terms in A168421, then the twin prime conjecture can be proved. - _John W. Nicholson_, Dec 05 2013
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A005382/b005382.txt">Table of n, a(n) for n = 1..10000</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, p. 870.
%H R. P. Boas & N. J. A. Sloane, <a href="/A005381/a005381.pdf">Correspondence, 1974</a>
%H Ajeet Kumar, Subhamoy Maitra, and Chandra Sekhar Mukherjee, <a href="https://doi.org/10.1007/s12095-020-00468-6">On approximate real mutually unbiased bases in square dimension</a>, Cryptography and Communications (2020) Vol. 13, 321-329.
%H Marius Tărnăuceanu, <a href="https://arxiv.org/abs/2003.10060">Arithmetic progressions in finite groups</a>, arXiv:2003.10060 [math.GR], 2020.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Cunningham_chain">Cunningham chain</a>
%F a(n) = A129521(n) / A005383(n). - _Reinhard Zumkeller_, Apr 19 2007
%F a(n) = (A005383(n) + 1)/2. - _Zak Seidov_, Nov 04 2010
%p f := proc(Q) local t1,i,j; t1 := []; for i from 1 to 500 do j := ithprime(i); if isprime(2*j-Q) then t1 := [op(t1),j]; fi; od: t1; end; f(1);
%t Select[Prime[Range[300]], PrimeQ[2#-1]&]
%o (Magma) [n: n in [0..1000] | IsPrime(n) and IsPrime(2*n-1)] // _Vincenzo Librandi_, Nov 18 2010
%o (PARI) select(p->isprime(2*p-1),primes(500)) \\ _Charles R Greathouse IV_, Apr 26 2012
%o (Haskell)
%o a005382 n = a005382_list !! (n-1)
%o a005382_list = filter
%o ((== 1) . a010051 . (subtract 1) . (* 2)) a000040_list
%o -- _Reinhard Zumkeller_, Oct 03 2012
%o (PARI) forprime(n=2, 10^3, if(ispseudoprime(2*n-1), print1(n, ", "))) \\ _Felix Fröhlich_, Jun 15 2014
%Y Cf. A005383, A005384 (2p+1), A057326, A057327, A057328, A057329, A057330, A005603, A063908 (2p-3), A063909 (2p-5), A023204 (2p+3), A000384, A001358.
%Y Cf. A010051, A000040, A053685 (subsequence), A006254.
%K nonn,easy
%O 1,1
%A _N. J. A. Sloane_
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