A088879 -id:A088879 - cp's OEIS frontend

A003627 Primes of the form 3n-1.

2, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401, 419, 431, 443, 449, 461, 467, 479, 491, 503, 509, 521, 557, 563, 569, 587

Offset: 1

N. J. A. Sloane and Mira Bernstein

Inert rational primes in the field Q(sqrt(-3)). - N. J. A. Sloane, Dec 25 2017
Primes p such that 1+x+x^2 is irreducible over GF(p). - Joerg Arndt, Aug 10 2011
Primes p dividing sum(k=0,p,C(2k,k)) -1 = A006134(p)-1. - Benoit Cloitre, Feb 08 2003
A039701(A049084(a(n))) = 2; A134323(A049084(a(n))) = -1. - Reinhard Zumkeller, Oct 21 2007
The set of primes of the form 3n - 1 is a superset of the set of lesser of twin primes larger than three (A001359). - Paul Muljadi, Jun 05 2008
Primes of this form do not occur in or as divisors of {n^2+n+1}. See A002383 (n^2+n+1 = prime), A162471 (prime divisors of n^2+n+1 not in A002383), and A002061 (numbers of the form n^2-n+1). - Daniel Tisdale, Jul 04 2009
Or, primes not in A007645. A003627 UNION A007645 = A000040. Also, primes of the form 6*k-5/2-+3/2. - Juri-Stepan Gerasimov, Jan 28 2010
Except for first term "2", all these prime numbers are of the form: 6*n-1. - Vladimir Joseph Stephan Orlovsky, Jul 13 2011
A088534(a(n)) = 0. - Reinhard Zumkeller, Oct 30 2011
For n>1: Numbers k such that (k-4)! mod k =(-1)^(floor(k/3)+1)*floor((k+1)/6), k>4. - Gary Detlefs, Jan 02 2012
Binomial(a(n),3)/a(n)= (3*A024893(n)^2+A024893(n))/2, n>1. - Gary Detlefs, May 06 2012
For every prime p in this sequence, 3 is a 9th power mod p. See Williams link. - Michel Marcus, Nov 12 2017
2 adjoined to A007528. - David A. Corneth, Nov 12 2017
For n >= 2 there exists a polygonal number P_s(3) = 3s - 3 = a(n) + 1. These are the only primes p with P_s(k) = p + 1, s >= 3, k >= 3, since P_s(k) - 1 is composite for k > 3. - Ralf Steiner, May 17 2018

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
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..1000
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].
A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Eric Weisstein's World of Mathematics, Eisenstein Prime
Kenneth S. Williams, 3 as a Ninth Power (mod p), Math. Scand., Vol 35 (1974), 309-317.
Index to sequences related to decomposition of primes in quadratic fields

Primes of form 3n+1 give A002476.
These are the primes arising in A024893, A087370, A088879. A091177 gives prime index.
Cf. A001359, A007528, A007645, A221717, A057145.
Subsequence of A034020.

a003627 n = a003627_list !! (n-1)
a003627_list = filter ((== 2) . (`mod` 3)) a000040_list
-- Reinhard Zumkeller, Oct 30 2011

[n: n in PrimesUpTo(720) | n mod 3 eq 2]; // Bruno Berselli, Apr 05 2011

t1 := {}; for n from 0 to 500 do if isprime(3*n+2) then t1 := {op(t1),3*n+2}; fi; od: A003627 := convert(t1,list);

Select[Range[-1, 600, 3], PrimeQ[#] &] (* Vincenzo Librandi, Jun 17 2015 *)
Select[Prime[Range[200]],Mod[#,3]==2&] (* Harvey P. Dale, Jan 31 2023 *)

is(n)=n%3==2 && isprime(n) \\ Charles R Greathouse IV, Mar 20 2013

From R. J. Mathar, Apr 03 2011: (Start)
Sum_{n>=1} 1/a(n)^2 = 0.30792... = A085548 - 1/9 - A175644.
Sum_{n>=1} 1/a(n)^3 = 0.134125... = A085541 - 1/27 - A175645. (End)

A024893 Numbers k such that 3*k+2 is prime.

Original entry on oeis.org

0, 1, 3, 5, 7, 9, 13, 15, 17, 19, 23, 27, 29, 33, 35, 37, 43, 45, 49, 55, 57, 59, 63, 65, 75, 77, 79, 83, 85, 87, 89, 93, 97, 103, 105, 115, 117, 119, 127, 129, 133, 139, 143, 147, 149, 153, 155, 159, 163, 167, 169, 173, 185, 187, 189, 195, 197, 199, 205, 213, 215, 217, 219, 225, 227

Offset: 1

Clark Kimberling

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

Cf. A003627 (associated primes), A091177 (gives prime index).

Cf. A087370, A088879.

[n: n in [0..1000] | IsPrime(3*n+2)]; // Vincenzo Librandi, Nov 20 2010

Select[Range[0, 250], PrimeQ[3# + 2] &] (* Vincenzo Librandi, Sep 25 2012 *)

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

a(n) = A087370(n)-1 = A088879(n)+1.

A087370 Numbers n such that 3n - 1 is a prime.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 14, 16, 18, 20, 24, 28, 30, 34, 36, 38, 44, 46, 50, 56, 58, 60, 64, 66, 76, 78, 80, 84, 86, 88, 90, 94, 98, 104, 106, 116, 118, 120, 128, 130, 134, 140, 144, 148, 150, 154, 156, 160, 164, 168, 170, 174, 186, 188, 190, 196, 198, 200, 206, 214, 216

Offset: 1

Giovanni Teofilatto, Oct 21 2003

3*n - 1 is an Eisenstein prime. - Vincenzo Librandi, Aug 08 2010

For all elements of this sequence there are no pairs (x,y) of positive integers such that a(n) = 3*x*y - x + y. - Pedro Caceres, Jan 28 2021

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

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

A003627 gives primes, A091177 gives prime index.

Cf. A010051, subsequence of A016789, A259645.

a087370 n = a087370_list !! (n-1)
a087370_list = filter ((== 1) . a010051' . subtract 1 . (* 3)) [0..]
-- Reinhard Zumkeller, Jul 03 2015

a(n)= A024893(n) + 1 = A088879(n) + 2.

Corrected and extended by Ray Chandler, Oct 22 2003

A091177 Numbers m such that the m-th prime is of the form 3*k-1.

Original entry on oeis.org

1, 3, 5, 7, 9, 10, 13, 15, 16, 17, 20, 23, 24, 26, 28, 30, 32, 33, 35, 39, 40, 41, 43, 45, 49, 51, 52, 54, 55, 56, 57, 60, 62, 64, 66, 69, 71, 72, 76, 77, 79, 81, 83, 86, 87, 89, 91, 92, 94, 96, 97, 98, 102, 103, 104, 107, 108, 109, 113, 116, 118, 119, 120, 123

Offset: 1

Ray Chandler, Dec 26 2003

A003627 indexed by A000040.

The asymptotic density of this sequence is 1/2 (by Dirichlet's theorem). - Amiram Eldar, Feb 28 2021

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A003627 (primes of the form 3*k-1), A024893, A087370, A088879.

A133677 is another version.

PrimePi/@Select[3Range[0,250]-1,PrimeQ]  (* Harvey P. Dale, Apr 26 2011 *)
Select[Range[150],IntegerQ[(Prime[#]+1)/3]&] (* Harvey P. Dale, Dec 14 2021 *)

a091177(limit)={my(m=0);forprime(p=2,prime(limit),m++;if(p%3==2,print1(m,", ")))};
a091177(123) \\ Hugo Pfoertner, Aug 03 2021

a(n) = k such that A000040(k) = A003627(n).

A107303 Numbers k such that (3*k - 5) is prime.

Original entry on oeis.org

4, 6, 8, 12, 14, 16, 22, 24, 26, 28, 34, 36, 38, 44, 48, 52, 54, 56, 62, 66, 68, 72, 76, 78, 82, 92, 94, 96, 104, 106, 112, 114, 118, 124, 126, 128, 134, 138, 142, 146, 148, 154, 156, 164, 168, 176, 182, 184, 192, 194, 202, 204, 206, 208, 212, 216, 222, 226, 232

Offset: 1

Parthasarathy Nambi, May 20 2005

3 and 5 are twin primes.

			If k=4, then 3*k - 5 = 7 (prime).
If k=28, then 3*k - 5 = 79 (prime).

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A088879.
Cf. A153183, A024892, A034936.
Equals A153183(n) + 1; also A024892(n) + 2; also A034936(n) + 3;

[n: n in [2..300] | IsPrime(3*n-5)]; // Vincenzo Librandi, Nov 13 2010

Select[Range[2,250],PrimeQ[3#-5]&] (* Harvey P. Dale, Mar 31 2024 *)

is(n)=isprime((3*n-5)) \\ Charles R Greathouse IV, Jun 12 2017

A108110 Numbers n such that prime(k)*n+prime(k+1), for k=1,...,6 all are primes.

Original entry on oeis.org

284, 3074, 3494, 21698, 32138, 43874, 51794, 60674, 75494, 407348, 437438, 459794, 571478, 660878, 667358, 705464, 716624, 740774, 811028, 820154, 910664, 1059398, 1077998, 1122584, 1150748, 1210754, 1222898, 1265018, 1412174, 1461164, 1486574, 1585868, 1631438

Offset: 1

Zak Seidov, Jun 03 2005

nonn

n == 0 (mod 2). n == 2 (mod 3). n == 3 or 4 (mod 5). - Jason Yuen, Sep 02 2024

			284 is OK because 2*284+3, 3*284+5, 5*284+7, 7*284+11, 11*284+13 and 13*284+17 all are primes.

Cf. A067076 (k=1), A088879 (k=2), A111224 (k=3), A101123 (k=4), A102721 (k=5).

Cf. A108117 (k=1..7), A379427 (k=1..8).

s={};Do[If[Union[PrimeQ/@Table[Prime[k]*n+Prime[k+1], {k, 6}]]=={True}, s=Append[s, n]], {n, 2, 1000000, 2}];s

\\ See isok from A108117
for(n=1,2*10^6,if(isok(n,6),print1(n", "))) \\ Jason Yuen, Sep 02 2024

a(22)-a(33) from Jason Yuen, Sep 02 2024

A108117 Numbers n such that prime(k)*n+prime(k+1), for k=1,...,7 all are primes.

Original entry on oeis.org

3494, 60674, 75494, 1122584, 2136044, 2473934, 3367244, 5600384, 6629804, 6910784, 7554644, 8572904, 10079144, 11848094, 11892164, 12043214, 12167594, 12269234, 12507284, 12700154, 13459664, 13924544, 14495354, 15005954, 16890914, 17827094, 20642984, 25796054

Offset: 1

Zak Seidov, Jun 03 2005

nonn

The only n, for which also 19*3494+23 is prime, is n=5600384. In principle, n == 4 (mod 10) can give primes of the form prime(k)*n+prime(k+1), for all k from 1 up to 41, while prime(42)*4+prime(43)=181*4+191 == 5 (mod 10) that is nonprime. It'd be very interesting to find at least one n such that prime(k)*n+prime(k+1), k=1,...,41 are all prime.

There are no values of n such that prime(k)*n+prime(k+1), k=1,...,9 are all prime. Proof: If n = 3*i then 2*(3*i)+3 = 3*(2*i+1) is not prime. If n = 3*i+1 then 5*(3*i+1)+7 = 3*(5*i+4) is not prime. If n = 3*i+2 then 23*(3*i+2)+29 = 3*(23*i+25) is not prime. - Jason Yuen, Sep 02 2024

			3494 is OK because 2*3494+3, 3*3494+5, 5*3494+7, 7*3494+11, 11*3494+13, 13*3494+17 and 17*3494+19 all are primes.

Cf. A067076 (k=1), A088879 (k=2), A111224 (k=3), A101123 (k=4), A102721 (k=5), A108976 (k=7).

Cf. A108110 (k=1..6), A379427 (k=1..8).

s={};Do[If[Union[PrimeQ/@Table[Prime[k]*n+Prime[k+1], {k, 7}]]=={True}, s=Append[s, n]], {n, 4, 10000000, 10}];s
Select[Range[9*10^6],AllTrue[Prime[Range[7]]#+Prime[Range[2,8]],PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 24 2018 *)

isok(n,upto=7)=for(k=1,upto,if(!isprime(prime(k)*n+prime(k+1)),return(0)));1
for(n=1,3*10^7,if(isok(n),print1(n", "))) \\ Jason Yuen, Sep 02 2024

a(13)-a(28) from Jason Yuen, Sep 02 2024

A124853 Numbers k such that 5k + 3 and 3k + 5 are primes.

Original entry on oeis.org

0, 2, 4, 8, 14, 16, 22, 32, 34, 44, 56, 58, 62, 74, 76, 86, 88, 92, 104, 118, 128, 146, 148, 154, 172, 196, 212, 218, 224, 232, 238, 256, 274, 284, 286, 308, 316, 322, 338, 364, 382, 386, 394, 428, 454, 476, 478, 494, 518, 526, 532, 536, 538, 568, 632, 664, 674

Offset: 1

Zak Seidov, Nov 10 2006

nonn

Intersection of A087505 and A088879.

All terms must be even. - Harvey P. Dale, Feb 12 2022

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A087505, A088879, A124854.

[n: n in [0..700] | IsPrime(5*n+3) and IsPrime(3*n+5)] // Vincenzo Librandi, Mar 26 2010

Select[Range[0,700,2],AllTrue[{5#+3,3#+5},PrimeQ]&] (* Harvey P. Dale, Feb 12 2022 *)

A124854 Primes p=n/2 such that 5n+3 and 3n+5 are primes.

Original entry on oeis.org

2, 7, 11, 17, 29, 31, 37, 43, 59, 73, 109, 137, 191, 193, 197, 227, 239, 263, 269, 337, 367, 373, 401, 409, 449, 479, 499, 541, 557, 701, 743, 757, 823, 827, 857, 941, 983, 997, 1033, 1051, 1109, 1163, 1303, 1361, 1471

Offset: 1

Zak Seidov, Nov 10 2006

nonn

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

Cf. A087505, A088879, A124853.

select(t -> isprime(t) and isprime(10*t+3) and isprime(6*t+5), [2,seq(i,i=3..10^4,2)]); # Robert Israel, Jun 11 2018

Select[Prime@ Range@ 250, AllTrue[{10 # + 3, 6 # + 5}, PrimeQ] &] (* Michael De Vlieger, Jun 11 2018 *)

isok(n) = isprime(n) && isprime(10*n+3) && isprime(6*n+5); \\ Michel Marcus, Oct 11 2013

cp's OEIS Frontend

A003627 Primes of the form 3n-1.

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A024893 Numbers k such that 3*k+2 is prime.

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A087370 Numbers n such that 3n - 1 is a prime.

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A091177 Numbers m such that the m-th prime is of the form 3*k-1.

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A107303 Numbers k such that (3*k - 5) is prime.

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A108110 Numbers n such that prime(k)*n+prime(k+1), for k=1,...,6 all are primes.

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A108117 Numbers n such that prime(k)*n+prime(k+1), for k=1,...,7 all are primes.

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