A063908 -id:A063908 - 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

A088878 Prime numbers p such that 3p - 2 is a prime.

Original entry on oeis.org

3, 5, 7, 11, 13, 23, 37, 43, 47, 53, 61, 67, 71, 103, 113, 127, 137, 163, 167, 181, 191, 193, 211, 251, 257, 263, 271, 277, 293, 307, 313, 331, 337, 347, 373, 401, 431, 433, 443, 461, 467, 487, 491, 523, 541, 557, 587, 593, 601, 673, 677, 727, 751, 757, 761

Offset: 1

Giovanni Teofilatto, Nov 27 2003

Indices of semiprime octagonal numbers. - Jonathan Vos Post, Feb 16 2006
Daughter primes of order 1. - Artur Jasinski, Dec 12 2007
A010051(3*a(n)-2) = 1. - Reinhard Zumkeller, Jul 02 2015

			For p = 3, 3p - 2 = 7;
for p = 523, 3p - 2 = 1567.

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta, UTET, CittaStudiEdizioni, Milano 1997

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Octagonal Number.
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A000040, A000567, A001222, A001358, A091179, A091180, A091181, A136019, A136020, A153183, A153184.
Cf. A010051, A063908, A172287.
Cf. A259730.

a088878 n = a088878_list !! (n-1)
a088878_list = filter ((== 1) . a010051' . subtract 2 . (* 3)) a000040_list
-- Reinhard Zumkeller, Jul 02 2015

[ p: p in PrimesUpTo(770) | IsPrime(3*p-2) ]; // Klaus Brockhaus, Dec 21 2008

lst={};Do[p=Prime[n];If[PrimeQ[3*p-2],AppendTo[lst,p]],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 22 2008 *)
n = 1; a = {}; Do[If[PrimeQ[(Prime[k] + 2n)/(2n + 1)], AppendTo[a, (Prime[k] + 2n)/(2n + 1)]], {k, 1, 500}]; a (* Artur Jasinski, Dec 12 2007 *)
Select[Prime[Range[150]],PrimeQ[3#-2]&] (* Harvey P. Dale, Feb 27 2024 *)

list(lim)=select(p->isprime(3*p-2),primes(primepi(lim))) \\ Charles R Greathouse IV, Jul 25 2011

Corrected and extended by Ray Chandler, Dec 27 2003

Entry revised by N. J. A. Sloane, Nov 28 2006, Jul 08 2010

A098090 Numbers k such that 2k-3 is prime.

Original entry on oeis.org

3, 4, 5, 7, 8, 10, 11, 13, 16, 17, 20, 22, 23, 25, 28, 31, 32, 35, 37, 38, 41, 43, 46, 50, 52, 53, 55, 56, 58, 65, 67, 70, 71, 76, 77, 80, 83, 85, 88, 91, 92, 97, 98, 100, 101, 107, 113, 115, 116, 118, 121, 122, 127, 130, 133, 136, 137, 140, 142, 143, 148, 155, 157, 158

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Sep 14 2004

Supersequence of A063908.

Left edge of the triangle in A065305. - Reinhard Zumkeller, Jan 30 2012

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000040, A085090, A086801.
Numbers n such that 2n+k is prime: A005097 (k=1), A067076 (k=3), A089038 (k=5), A105760 (k=7), A155722 (k=9), A101448 (k=11), A153081 (k=13), A089559 (k=15), A173059 (k=17), A153143 (k=19).
Numbers n such that 2n-k is prime: A006254 (k=1), this sequence (k=3), A089253 (k=5), A089192 (k=7), A097069 (k=9), A097338 (k=11), A097363 (k=13), A097480 (k=15), A098605 (k=17), A097932 (k=19).

Filtered([1..200], k-> IsPrime(2*k-3) ); # G. C. Greubel, May 21 2019

[ n: n in [1..200] | IsPrime(2*n-3) ]; // Vincenzo Librandi, Dec 26 2010

(Prime[Range[2,100]]+3)/2 (* Vladimir Joseph Stephan Orlovsky, Feb 08 2010 *)
Select[Range[200],PrimeQ[2#-3]&] (* Harvey P. Dale, Mar 05 2022 *)

is(n)=isprime(2*n-3) \\ Charles R Greathouse IV, Feb 17 2017

[n for n in (1..200) if is_prime(2*n-3) ] # G. C. Greubel, May 21 2019

Half of p + 3, where p is a prime greater than 2.

A122845(a(n), 3) = 3; a(n) = A113935(n+1)/2. - Reinhard Zumkeller, Sep 14 2006

A092109 Primes p such that p+3 is a semiprime.

Original entry on oeis.org

3, 7, 11, 19, 23, 31, 43, 59, 71, 79, 83, 103, 131, 139, 163, 191, 199, 211, 223, 251, 271, 311, 331, 359, 379, 383, 419, 443, 463, 479, 499, 523, 563, 619, 631, 659, 691, 743, 839, 859, 863, 883, 911, 919, 971, 1039, 1091, 1123, 1151, 1171, 1223, 1231, 1259

Offset: 1

Zak Seidov, Feb 21 2004

Primes p such that p-3 is semiprime are in A089531; p and 2p+3 both prime, A023204; p, 2p-3 and 2p+3 prime, A092110.
Primes p such that (p+3)/2 is prime. All these primes are congruent to 3 mod 4. - Artur Jasinski, Oct 11 2008
Subsequence of A131426. - Zak Seidov, Mar 29 2015
Subsequence of A091305. - David Radcliffe, May 22 2022

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

Cf. A023204, A089531, A063908, A092110, A115334, A131426.

IsSemiprime:=func< p | &+[ k[2]: k in Factorization(p)] eq 2 >; [p: p in PrimesUpTo(1300)| IsSemiprime(p+3)]; // Vincenzo Librandi, Feb 21 2014

select(p -> isprime(p) and isprime((p+3)/2), [seq(2*k+1,k=1..1000)]); # Robert Israel, Mar 29 2015

aa = {}; k = 3; Do[If[PrimeQ[(k + Prime[n])/2], AppendTo[aa, Prime[n]]], {n, 1, 100}]; aa (* Artur Jasinski, Oct 11 2008 *)
Select[Prime[Range[300]],PrimeOmega[#+3]==2&] (* Harvey P. Dale, Feb 07 2018 *)

is(n)=n%2 && isprime((n+3)/2) && isprime(n) \\ Charles R Greathouse IV, Jul 12 2016

a(n) = 2*A063908(n)-3 = 4*A115334(n)+3. - Artur Jasinski, Oct 11 2008

A145490 Numbers k such that 6k+19 is prime and absolute value of 12k+1 is also prime.

Original entry on oeis.org

-2, -1, 3, 8, 9, 13, 15, 20, 23, 29, 34, 35, 48, 55, 59, 63, 69, 73, 78, 84, 93, 100, 104, 115, 119, 134, 135, 139, 148, 150, 169, 174, 178, 185, 189, 199, 203, 210, 213, 218, 238, 254, 255, 260, 265, 268, 275, 280, 288, 289, 293, 294, 295, 308, 309, 335, 344

Offset: 1

Artur Jasinski, Oct 11 2008

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A063908-A063913, A092109, A145471-A145490.

with(numtheory): a:=proc(n) if(isprime(6*n+19) and isprime(abs(12*n+1)))then return n: fi: return NULL: end: seq(a(n),n=-2..350); # Nathaniel Johnston, Jul 26 2011

a(n) = (A145480(n-2)-1)/12 for n >= 3.

Corrected by Arkadiusz Wesolowski, Jul 26 2011

A259730 Primes p such that both 2p - 3 and 3p - 2 are prime.

Original entry on oeis.org

3, 5, 7, 11, 13, 23, 37, 43, 53, 67, 71, 113, 127, 137, 167, 181, 191, 193, 211, 251, 263, 331, 347, 373, 431, 433, 443, 461, 487, 587, 727, 751, 757, 907, 991, 1021, 1091, 1103, 1171, 1187, 1213, 1231, 1297, 1367, 1453, 1483, 1597, 1637, 1663, 1667, 1733

Offset: 1

Reinhard Zumkeller, Jul 05 2015

nonn

A010051(2*a(n) - 3) * A010051(3*a(n) - 2) = 1;
A259758(n) = (2*a(n) - 3) * (3*a(n) - 2).
Except for a(1)=3 this is the same sequence as primes p such that A288814(3*p) - A288814(2*p) = 5. - David James Sycamore, Jul 22 2017
Furthermore, (A288814(3*p)*A288814(2*p))/6 belongs to A259758. - David James Sycamore, Jul 23 2017

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Intersection of A063908 and A088878; A172287, A259758.

Cf. A288814, A259758.

import Data.List.Ordered (isect)
a259730 n = a259730_list !! (n-1)
a259730_list = a063908_list `isect` a088878_list

Select[Prime@ Range@ 270, Times @@ Boole@ Map[PrimeQ, {2 # - 3, 3 # - 2}] > 0 &] (* Michael De Vlieger, Jul 22 2017 *)
Select[Prime[Range[300]],AllTrue[{2#-3,3#-2},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 08 2020 *)

lista(nn) = forprime(p=3, nn, if(isprime(2*p-3) && isprime(3*p-2), print1(p, ", "))); \\ Altug Alkan, Jul 22 2017

A092110 Primes p such that 2p+3 and 2p-3 are both prime.

Original entry on oeis.org

5, 7, 13, 17, 43, 53, 67, 97, 113, 127, 137, 157, 167, 193, 223, 283, 487, 547, 563, 613, 617, 643, 647, 743, 773, 937, 1033, 1187, 1193, 1277, 1427, 1453, 1483, 1543, 1583, 1627, 1663, 1733, 1847, 2027, 2143, 2297, 2393, 2437, 2467, 2477, 2503, 2617, 2843

Offset: 1

Zak Seidov, Feb 21 2004

Intersection of A023204 and A063908.

All numbers in this sequence end with 3 or 7 (except the first one, which is 5). See A136191 or A136192. - Carlos Alves, Dec 20 2007

			From _K. D. Bajpai_, Sep 08 2020: (Start)
7 is a term because 2*7 + 3 = 17 and 2*7 - 3 = 11 are both prime.
13 is a term because 2*13 + 3 = 29 and 2*13 - 3 = 23 are both prime.
(End)

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A023204, A089531, A063908, A092109.

Cf. A136191, A136192.

[p: p in PrimesUpTo(10000)|IsPrime(2*p-3) and IsPrime(2*p+3)] // Vincenzo Librandi, Nov 16 2010

select(p -> isprime(p) and isprime(2*p+3) and isprime(2*p-3), [seq(2*k+1, k=1..1000)]); # K. D. Bajpai, Sep 08 2020

Select[Prime@Range@1000,PrimeQ[2#-3]&&PrimeQ[2#+3]&] (* Vladimir Joseph Stephan Orlovsky, Apr 25 2011 *)

A172287 Primes p such that exactly one of 2p-3 and 3p-2 is prime.

Original entry on oeis.org

17, 31, 41, 47, 61, 83, 97, 101, 103, 107, 157, 163, 223, 233, 241, 257, 271, 277, 283, 293, 307, 311, 313, 317, 337, 401, 421, 457, 467, 491, 521, 523, 541, 547, 557, 563, 577, 593, 601, 613, 617, 631, 641, 643, 647, 653, 661, 673, 677, 701, 743, 761, 773

Offset: 1

Juri-Stepan Gerasimov, Jan 30 2010

A010051(2*a(n)+3) + A010051(3*a(n)+2) = 1; each term is either a term of A063908 or of A088878. - Reinhard Zumkeller, Jul 02 2015

No terms end in 9. Dickson's conjecture implies that there are infinitely many terms. - Robert Israel, Jul 02 2015

			a(1)=17 because 2*17-3=31 is prime and 3*17-2=49 is nonprime.
19 is not a term because neither 2*19-3=35 nor 3*19-2=55 is prime;
23 is not a term because both 2*23-3=43 and 3*23-2=67 are prime.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040, A010051, A063908, A088878, A259730.

a172287 n = a172287_list !! (n-1)
a172287_list = filter
   (\p -> a010051' (2 * p - 3) + a010051' (3 * p - 2) == 1) a000040_list
-- Reinhard Zumkeller, Jul 02 2015

A172287:=n->`if`(isprime(n) and (isprime(2*n-3) xor isprime(3*n-2)), n, NULL): seq(A172287(n), n=1..1000); # Wesley Ivan Hurt, Jun 23 2015

Select[Prime@ Range@ 150, Xor[PrimeQ[2 # - 3], PrimeQ[3 # - 2]] &] (* Michael De Vlieger, Jul 01 2015 *)

Extended by Charles R Greathouse IV, Mar 25 2010

A171517 Primes p such that 2*p+11 is prime.

Original entry on oeis.org

3, 13, 31, 43, 73, 109, 151, 163, 181, 193, 199, 211, 223, 283, 331, 349, 373, 379, 409, 421, 433, 463, 499, 541, 571, 601, 613, 619, 643, 709, 739, 769, 823, 829, 883, 991, 1009, 1021, 1039, 1051, 1063, 1129, 1213, 1231, 1291, 1303, 1423, 1453, 1471, 1549

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 10 2009

			2*3+11=17, which is prime.

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

Cf. A000040, A005382, A005384, A023204 - A023207, A063908 - A063912.

[p: p in PrimesUpTo(1600) | IsPrime(2*p+11)]; // Vincenzo Librandi, Apr 27 2014

Select[Prime[Range[6! ]],PrimeQ[2*#+11]&]

A145487 Numbers k such that 6k+5 is prime and 12k+5 is also prime.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 8, 9, 11, 14, 16, 21, 22, 24, 29, 32, 37, 38, 42, 43, 46, 51, 58, 63, 64, 66, 71, 73, 77, 79, 81, 84, 92, 98, 99, 102, 106, 107, 108, 113, 119, 123, 134, 136, 142, 143, 156, 157, 158, 162, 184, 191, 196, 198, 203, 212, 217, 219, 227, 228, 238, 241, 246

Offset: 1

Artur Jasinski, Oct 11 2008

nonn

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A063908-A063913, A092109, A145471-A145490.

aa = {}; k = 5; Do[If[PrimeQ[(k + Prime[n])/2], AppendTo[aa, (Prime[n] - 5)/12]], {n, 1, 500}]; aa
Select[Range[0, 250], PrimeQ[6 # + 5] && PrimeQ[12 # + 5] &] (* Ivan Neretin, Jan 21 2017 *)
Select[Range[0,250],AllTrue[5+{6#,12#},PrimeQ]&] (* Harvey P. Dale, Dec 20 2022 *)

a(n) = (A145471(n)-5)/12.

cp's OEIS Frontend

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

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A088878 Prime numbers p such that 3p - 2 is a prime.

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A098090 Numbers k such that 2k-3 is prime.

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A092109 Primes p such that p+3 is a semiprime.

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A145490 Numbers k such that 6k+19 is prime and absolute value of 12k+1 is also prime.

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Maple

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A259730 Primes p such that both 2*p - 3 and 3*p - 2 are prime.

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Haskell

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A259730 Primes p such that both 2p - 3 and 3p - 2 are prime.