A063909 -id:A063909 - cp's OEIS frontend

A005382 Primes p such that 2p-1 is also prime.

2, 3, 7, 19, 31, 37, 79, 97, 139, 157, 199, 211, 229, 271, 307, 331, 337, 367, 379, 439, 499, 547, 577, 601, 607, 619, 661, 691, 727, 811, 829, 877, 937, 967, 997, 1009, 1069, 1171, 1237, 1279, 1297, 1399, 1429, 1459, 1531, 1609, 1627, 1657, 1759, 1867, 2011

Offset: 1

N. J. A. Sloane

Sequence gives values of p such Sum_{i=1..p} gcd(p,i) = A018804(p) is prime. - Benoit Cloitre, Jan 25 2002
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
Primes in A006254. - Zak Seidov, Mar 26 2013
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
Records subsequence of A023509 (n >= 2). - David James Sycamore, May 05 2025

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

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, p. 870.
R. P. Boas and N. J. A. Sloane, Correspondence, 1974
Ajeet Kumar, Subhamoy Maitra, and Chandra Sekhar Mukherjee, On approximate real mutually unbiased bases in square dimension, Cryptography and Communications (2020) Vol. 13, 321-329.
Sagar Mandal, Divisibility and Sequence Properties of sigma+ and phi+, arXiv:2508.11660 [math.GM], 2025. See p. 8.
Marius Tărnăuceanu, Arithmetic progressions in finite groups, arXiv:2003.10060 [math.GR], 2020.
Wikipedia, Cunningham chain

Cf. A005383, A005384 (2p+1), A057326, A057327, A057328, A057329, A057330, A005603, A063908 (2p-3), A063909 (2p-5), A023204 (2p+3), A000384, A001358.
Cf. A010051, A000040, A053685 (subsequence), A006254.
Cf. A023509.

a005382 n = a005382_list !! (n-1)
a005382_list = filter
   ((== 1) . a010051 . (subtract 1) . (* 2)) a000040_list
-- Reinhard Zumkeller, Oct 03 2012

[n: n in [0..1000] | IsPrime(n) and IsPrime(2*n-1)]; // Vincenzo Librandi, Nov 18 2010

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);
# second Maple program:
q:= p-> andmap(isprime, [p, 2*p-1]):
select(q, [$2..2500])[];  # Alois P. Heinz, Dec 16 2024

Select[Prime[Range[300]], PrimeQ[2#-1]&]

select(p->isprime(2*p-1),primes(500)) \\ Charles R Greathouse IV, Apr 26 2012

forprime(n=2, 10^3, if(ispseudoprime(2*n-1), print1(n, ", "))) \\ Felix Fröhlich, Jun 15 2014

a(n) = A129521(n) / A005383(n). - Reinhard Zumkeller, Apr 19 2007

a(n) = (A005383(n) + 1)/2. - Zak Seidov, Nov 04 2010

A145471 Primes p such that (5+p)/2 is prime.

Original entry on oeis.org

5, 17, 29, 41, 53, 89, 101, 113, 137, 173, 197, 257, 269, 293, 353, 389, 449, 461, 509, 521, 557, 617, 701, 761, 773, 797, 857, 881, 929, 953, 977, 1013, 1109, 1181, 1193, 1229, 1277, 1289, 1301, 1361, 1433, 1481, 1613, 1637, 1709, 1721, 1877, 1889, 1901

Offset: 1

Artur Jasinski, Oct 11 2008

nonn

All these primes are congruent to 1 mod 4 and to 5 mod 12.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A063909, A092109.

Subsequence of A040117. - Zak Seidov, Feb 21 2016

[p: p in PrimesInInterval(3,2000) | IsPrime((5+p) div 2)]; // Vincenzo Librandi, Feb 25 2016

select(t -> isprime(t) and isprime((t+5)/2), [seq(i, i=5..1000, 12)]); # Robert Israel, Feb 24 2016

aa = {}; k = 5; Do[If[PrimeQ[(k + Prime[n])/2], AppendTo[aa, Prime[n]]], {n, 1, 500}];aa
Select[Prime[Range[500]],PrimeQ[(5+#)/2]&]  (* Harvey P. Dale, Apr 23 2011 *)

forprime(p=2,1e4,if(p%12!=5,next);if(isprime(p\2+3),print1(p", "))) \\ Charles R Greathouse IV, Jul 16 2011

a(n) = 2*A063909(n)-5. - Robert Israel, Feb 24 2016

A230138 List of those primes p with p + 2 and 2*p - 5 both prime.

Original entry on oeis.org

5, 11, 17, 29, 59, 71, 101, 137, 149, 179, 197, 227, 281, 311, 431, 599, 617, 641, 809, 821, 857, 1151, 1277, 1319, 1451, 1481, 1487, 1607, 1667, 1697, 1997, 2027, 2081, 2111, 2129, 2339, 2657, 2711, 2789, 3167, 3329, 3371, 3461, 3557, 3767, 3917, 3929, 4049, 4217, 4259

Offset: 1

Zhi-Wei Sun, Oct 10 2013

nonn

Clearly, all terms are congruent to 5 modulo 6, and not congruent to 3 modulo 5. Primes in this sequence are sparser than twin primes and Sophie Germain primes.
This sequence is interesting because of the conjectures in the comments in A230140 and A230141.
Intersection of A001359 and A089253 (or A063909). - M. F. Hasler, Oct 10 2013

			a(1) = 5 since neither 2 + 2 nor 2*3 - 5 is prime, but 5 + 2 = 7 and 2*5 - 5 = 5 are both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A001359, A006512, A005384, A230140, A230141.

PQ[p_]:=PQ[p]=PrimeQ[p+2]&&PrimeQ[2p-5]
m=0
Do[If[PQ[Prime[n]],m=m+1;Print[m," ",Prime[n]]],{n,1,584}]

is_A230138(p)=isprime(p)&&isprime(p+2)&&isprime(p*2-5) \\ For large p it might be much faster to check first whether p%6==5. - M. F. Hasler, Oct 10 2013

A290838 a(n) = smallest prime p such that 2p - 2n + 1 is prime.

Original entry on oeis.org

2, 2, 3, 5, 5, 7, 7, 13, 11, 11, 11, 13, 13, 19, 17, 17, 17, 19, 19, 37, 23, 23, 23, 29, 29, 31, 29, 29, 29, 31, 31, 37, 37, 41, 37, 37, 37, 43, 41, 41, 41, 43, 43, 61, 47, 47, 47, 53, 53, 67, 53, 53, 53, 59, 59, 61, 59, 59, 59, 61, 61, 67, 67, 71, 67, 67, 67, 73

Offset: 0

XU Pingya, Aug 11 2017

a(n) > n. - Iain Fox, Nov 13 2017

Iain Fox, Table of n, a(n) for n = 0..20000

Cf. A005382, A063908, A063909, A063910, A063911, A063912, A063913, A020483, A290839.

Table[j=0; found=False; While[!found,j++; found=PrimeQ[2Prime[j]-2n+1] && 2Prime[j]-2n+1>0]; Prime[j],{n,67}]
(* Second program: *)
Table[SelectFirst[Prime@ Range[n^2], And[# > 0, PrimeQ@ #] &[2 # - 2 n + 1] &], {n, 67}] (* Michael De Vlieger, Aug 14 2017 *)

a(n) = {my(p=2); while(!isprime(2*p-2*n+1), p = nextprime(p+1)); p; } \\ Michel Marcus, Aug 12 2017

a(n) = forprime(p=n+1, , if(isprime(2*p - 2*n + 1), return(p))) \\ Iain Fox, Nov 13 2017

a(-n) = A290839(n+1) - Iain Fox, Dec 14 2017

a(0) prepended by Iain Fox, Dec 14 2017

A167462 Primes p such that 2*p-5 is composite.

Original entry on oeis.org

7, 13, 19, 31, 37, 41, 43, 61, 67, 73, 79, 83, 97, 103, 107, 109, 113, 127, 139, 151, 157, 163, 167, 173, 181, 191, 193, 199, 211, 223, 229, 239, 241, 251, 269, 271, 277, 283, 293, 307, 313, 317, 331, 337, 347, 349, 359, 367, 373, 379, 397, 409, 419, 421

Offset: 1

Vincenzo Librandi, Nov 04 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A063909.

[p: p in PrimesInInterval(4,600)| not IsPrime(2*p-5)]; // Vincenzo Librandi, Sep 15 2013

Select[Range[4, 3000], PrimeQ[#]&& ! PrimeQ[2 # - 5] &](* Vincenzo Librandi, Sep 15 2013 *)
Select[Prime[Range[100]],CompositeQ[2#-5]&] (* Harvey P. Dale, May 16 2016 *)

Corrected (229 inserted) by R. J. Mathar, Nov 30 2009

Corrected (3 deleted, as 2*3-5 = 1 and 1 is not composite) by Harvey P. Dale, May 16 2016

cp's OEIS Frontend

A005382 Primes p such that 2p-1 is also prime.

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A145471 Primes p such that (5+p)/2 is prime.

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A230138 List of those primes p with p + 2 and 2*p - 5 both prime.

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A290838 a(n) = smallest prime p such that 2p - 2n + 1 is prime.

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A167462 Primes p such that 2*p-5 is composite.

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