A091177 -id:A091177 - 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)

A039701 a(n) = n-th prime modulo 3.

Original entry on oeis.org

2, 0, 2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 1

Offset: 1

Clark Kimberling

If n > 2 and prime(n) is a Mersenne prime then a(n) = 1. Proof: prime(n) = 2^p - 1 for some odd prime p, so prime(n) = 2*4^((p-1)/2) - 1 == 2 - 1 = 1 (mod 3). - Santi Spadaro, May 03 2002; corrected and simplified by Dean Hickerson, Apr 20 2003
Except for n = 2, a(n) is the smallest number k > 0 such that 3 divides prime(n)^k - 1. - T. D. Noe, Apr 17 2003
a(n) <> 0 for n <> 2; a(A049084(A003627(n))) = 2; a(A049084(A002476(n))) = 1; A134323(n) = (1 - 0^a(n)) * (-1)^(a(n)+1). - Reinhard Zumkeller, Oct 21 2007
Probability of finding 1 (or 2) in this sequence is 1/2. This follows from the Prime Number Theorem in arithmetic progressions. Examples: There are 4995 1's in terms (10^9 +1) to (10^9+10^4); there are 10^9/2-1926 1's in the first 10^9 terms. - Jerzy R Borysowicz, Mar 06 2022

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

Cf. A091178 (indices of 1's), A091177 (indices of 2's).
Cf. A120326 (partial sums).
Cf. A185934, A217659.
Cf. A010872.
Other moduli: A039702-A039706, A038194, A007652, A039709-A039715.

a039701 = (`mod` 3) . a000040
a039701_list = map (`mod` 3) a000040_list
-- Reinhard Zumkeller, Nov 16 2012

[p mod(3): p in PrimesUpTo(500)]; // Vincenzo Librandi, May 06 2014

seq(ithprime(n) mod 3, n=1..105); # Nathaniel Johnston, Jun 29 2011

Mathematica
```
Table[Mod[Prime[n], 3], {n, 100}]
```

primes(100)%3 \\ Charles R Greathouse IV, May 06 2014

Sum_k={1..n} a(k) ~ (3/2)*n. - Amiram Eldar, Dec 11 2024

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.

A091178 Numbers k such that k-th prime is of the form 6*m+1.

Original entry on oeis.org

4, 6, 8, 11, 12, 14, 18, 19, 21, 22, 25, 27, 29, 31, 34, 36, 37, 38, 42, 44, 46, 47, 48, 50, 53, 58, 59, 61, 63, 65, 67, 68, 70, 73, 74, 75, 78, 80, 82, 84, 85, 88, 90, 93, 95, 99, 100, 101, 105, 106, 110, 111, 112, 114, 115, 117, 121, 122, 125, 127, 129, 130

Offset: 1

Ray Chandler, Dec 26 2003

nonn

A002476 indexed by A000040.
Also k for which prime(k) == 1 (mod 3). - Bruno Berselli, Mar 04 2016
Sequence A091177 (indices of primes of the form 3*k-1) is this sequence's complement in the positive integers without {2}. - M. F. Hasler, Sep 02 2016
The asymptotic density of this sequence is 1/2 (by Dirichlet's theorem). - Amiram Eldar, Feb 28 2021

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A000040, A002476 (primes of the form 6*m+1), A091177 (indices of primes of the form 3*k-1), A024892, A024899.

Select[Range[200],IntegerQ[(Prime[#]-1)/6]&] (* Harvey P. Dale, Aug 25 2013 *)

isok(n) = !((prime(n)-1) % 6); \\ Michel Marcus, Mar 04 2016

a(n) = k such that A000040(k) = A002476(n).

Definition edited by Zak Seidov, Oct 09 2014

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

A088879 Numbers n such that 3n + 5 is a prime.

Original entry on oeis.org

-1, 0, 2, 4, 6, 8, 12, 14, 16, 18, 22, 26, 28, 32, 34, 36, 42, 44, 48, 54, 56, 58, 62, 64, 74, 76, 78, 82, 84, 86, 88, 92, 96, 102, 104, 114, 116, 118, 126, 128, 132, 138, 142, 146, 148, 152, 154, 158, 162, 166, 168, 172, 184, 186, 188, 194, 196, 198, 204, 212, 214, 216, 218

Offset: 1

Giovanni Teofilatto, Nov 27 2003

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

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

A003627 gives primes, A091177 gives prime index.

Cf. A016789.

lst={};Do[p=n+(n+2)+(n+3);If[PrimeQ[p],AppendTo[lst,n]],{n,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, May 22 2009 *)
Select[Range[-1,300],PrimeQ[3#+5]&] (* Harvey P. Dale, Jun 08 2020 *)

a(n) = A024893(n) - 1 = A087370(n) - 2.

Edited and extended by Ray Chandler, Dec 26 2003

A134323 a(n) = Legendre(-3, prime(n)).

Original entry on oeis.org

-1, 0, -1, 1, -1, 1, -1, 1, -1, -1, 1, 1, -1, 1, -1, -1, -1, 1, 1, -1, 1, 1, -1, -1, 1, -1, 1, -1, 1, -1, 1, -1, -1, 1, -1, 1, 1, 1, -1, -1, -1, 1, -1, 1, -1, 1, 1, 1, -1, 1, -1, -1, 1, -1, -1, -1, -1, 1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 1, -1, 1, -1, -1, 1, 1, 1, -1, -1, 1, -1, 1, -1, 1, -1, 1, 1, -1, -1, 1, -1, 1, -1, -1, 1, -1, 1, -1, -1, -1, 1, 1, 1, -1

Offset: 1

Reinhard Zumkeller, Oct 21 2007

Value of lowest trit of prime(n) in balanced ternary representation (A059095) (original definition).

For p = prime(n) != 3, a(n) = +1 if p is of the form 3*k + 1, and -1 if the p is of the form 3*k - 1. - Joerg Arndt, Sep 16 2014

			For n=20, prime(20) = 71, and we verify that -3 is not a quadratic residue modulo 71, hence a(20) = -1. Also, we see that the balanced ternary representation row A059095(71) = {1, 0, -1, 0, -1} which ends in -1.
For n=21, prime(21) = 73, and we see that x^2 = -3 mod 73 has solutions like x = 17, 56, hence a(21) = 1. Also, the balanced ternary representation row A059095(73) = {1, 0 -1, 0, 1} which ends in 1.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000040, A039701, A049084, A112632 (partial sums), A059095 (balanced ternary)
Cf. A091177 (indices of -1's), A091178 (indices of +1's), A003627, A002476.
Other moduli: A070750, A257834.
Cf. A102283.

a134323 n = (1 - 0 ^ m) * (-1) ^ (m + 1) where m = a000040 n `mod` 3
-- Reinhard Zumkeller, Sep 16 2014

A134323[n_] := (r = Mod[Prime[n], 6]; If[r == 1, 1, -1]); A134323[1] = -1; A134323[2] = 0; Table[A134323[n], {n, 1, 102}] (* Jean-François Alcover, Nov 07 2011, after Bill McEachen *)
JacobiSymbol[-3, Prime[Range[100]]] (* Alonso del Arte, Aug 02 2017 *)

apply(p->kronecker(-3,p), primes(100)) \\ Charles R Greathouse IV, Aug 14 2017

-1 if the n-th prime is 2 or == 5 mod 6, +1 if the n-th prime is == 1 mod 6, and 0 if it is 3.
a(n) = (1 - 0^A039701(n)) * (-1)^(A039701(n)+1).
a(n) != 0 for n != 2;
a(A049084(A003627(n))) = -1; a(A049084(A002476(n))) = +1.
a(n) = A102283(prime(n)). - Ridouane Oudra, Jan 09 2025

Wording of definition changed by N. J. A. Sloane, Jun 21 2015

Name simplified by Alonso del Arte, Aug 02 2017

A133677 Integers k such that prime(k)(2prime(k)-1)/3 is an integer.

Original entry on oeis.org

1, 2, 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, 124, 126

Offset: 1

Roger L. Bagula, Dec 28 2007

nonn

Apart from the term "2", the same as A091177. - Stefan Steinerberger, Dec 29 2007
Numbers n such that the number of distinct residues r in the congruence x^3 == r (mod p) is equal to p where p = prime(n). See A046530. - Michel Lagneau, Sep 28 2016
The asymptotic density of this sequence is 1/2 (by Dirichlet's theorem). - Amiram Eldar, Feb 28 2021

			4 is not in the sequence since prime(4)*(2*prime(4) - 1)/3 = 7*(2*7 - 1)/3 = 7*13/3 = 91/3 is not an integer, but 5 is in the sequence since prime(5)*(2*prime(5) - 1)/3 = 11*(2*11 - 1)/3 = 11*21/3 = 11*7 = 77 is an integer. - _Michael B. Porter_, Sep 28 2016

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

Cf. A046530, A091177, A133645.

for n from 1 to 126 do if((ithprime(n) mod 3) mod 2=0) then print(n) fi od; # Gary Detlefs, Dec 06 2011

Union[Table[If[IntegerQ[Prime[n]*(2*Prime[n] - 1)/3], n, {}], {n, 1, 100}]]
pnQ[n_]:=Module[{pn=Prime[n]},IntegerQ[(pn(2pn-1))/3]]; Select[Range[ 150], pnQ] (* Harvey P. Dale, Oct 02 2011 *)
Sort@ Join[{2}, Select[ Range@ 126, Mod[2*Prime[#], 3] == 1 &]] (* Robert G. Wilson v, Sep 28 2016 *)
Select[Range[126], IntegerQ[Prime[#]*(2 *Prime[#] - 1)/3] &] (* Robert Price, Apr 19 2025 *)

Integers k such that (prime(k) mod 3) mod 2 = 0. - Gary Detlefs, Dec 06 2011

A276414 Index of the first prime which starts a run of n consecutive primes all congruent to each other mod 3 (or mod 6).

Original entry on oeis.org

1, 9, 15, 54, 271, 271, 2209, 11199, 13717, 13717, 34369, 136456, 172146, 1254203, 1254203, 4308948, 12762142, 21357253, 25813464, 25813464, 39500857, 39500857, 947438659, 947438659, 947438659, 5703167678, 5703167678, 16976360924, 57446769091, 57446769091, 57446769091

Offset: 1

M. F. Hasler, Sep 02 2016

nonn

First term of runs of increasing length of consecutive integers in A270190. - M. F. Hasler, Sep 03 2016

			prime(9) = 23 starts the first run of 2 consecutive primes, {23, 29}, which are congruent to each other (mod 6). Therefore a(2) = 9.
prime(15) = 47 starts the first run of 3 consecutive primes, {47, 53, 59}, which are congruent to each other (mod 6). Therefore a(3) = 15.
prime(54) = 251 starts the first run of 4 consecutive primes, {251, 257, 263, 269}, which are congruent to each other (mod 6). Therefore a(4) = 54.
prime(271) = 1741 starts the first run of 5 consecutive primes, {1741, 1747, 1753, 1759, 1777}, which are congruent to each other (mod 6). Therefore a(5) = 271. This is the first case where the primes are of the form 3k+1.
prime(271) = 1741 also starts the first run of 6 consecutive primes, {1741, 1747, 1753, 1759, 1777, 1783}, which are all congruent to each other (mod 6). Therefore a(6) = 271, too.

J. K. Andersen, Consecutive Congruent Primes.

Cf. A054679; A247816, A247967; A091177, A091178; A000720, A000040.

Subsequence of A270190 (after discarding 1 and duplicates of other terms).

m=c=i=o=0;print1(1);forprime(p=1,,i++;(o-o=p)%3&&(!c||!c=0)&&next;c++>m||next;print1(",",i-m=c))

a(n) = A000720(A054679(n)).

a(30)-a(31) from and name clarified by Jinyuan Wang, Feb 24 2020

A171820 Numbers n such that the n-th prime is of the form 3k + 1/2 +- 1/2.

Original entry on oeis.org

2, 4, 6, 8, 11, 12, 14, 18, 19, 21, 22, 25, 27, 29, 31, 34, 36, 37, 38, 42, 44, 46, 47, 48, 50, 53, 58, 59, 61, 63, 65, 67, 68, 70, 73, 74, 75, 78, 80, 82, 84, 85, 88, 90, 93, 95, 99, 100, 101, 105, 106, 110, 111, 112, 114, 115, 117, 121, 122, 125, 127, 129, 130, 131, 133

Offset: 1

Juri-Stepan Gerasimov, Dec 19 2009

Numbers n such that the n-th prime is a generalized Cuban prime (A007645). A171820 UNION A091177 = A000027.

Essentially the same as A091178. - R. J. Mathar, Jan 28 2010

			2 is in the sequence because 2nd prime = 3*1 + 1/2 - 1/2;
4 is in the sequence because 4th prime = 3*2 + 1/2 + 1/2.

Cf. A000027, A000040, A007645, A091177.

list(lim)=my(v=List(),n); forprime(p=2,, if(n++>lim, return(Vec(v))); if(p%3<2, listput(v,n))) \\ Charles R Greathouse IV, Feb 07 2017

cp's OEIS Frontend

A003627 Primes of the form 3n-1.

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A039701 a(n) = n-th prime modulo 3.

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

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A091178 Numbers k such that k-th prime is of the form 6*m+1.

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

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A088879 Numbers n such that 3n + 5 is a prime.

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A134323 a(n) = Legendre(-3, prime(n)).

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A133677 Integers k such that prime(k)(2prime(k)-1)/3 is an integer.