A024892 -id:A024892 - cp's OEIS frontend

A002476 Primes of the form 6m + 1.

7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139, 151, 157, 163, 181, 193, 199, 211, 223, 229, 241, 271, 277, 283, 307, 313, 331, 337, 349, 367, 373, 379, 397, 409, 421, 433, 439, 457, 463, 487, 499, 523, 541, 547, 571, 577, 601, 607, 613, 619

Offset: 1

N. J. A. Sloane

Equivalently, primes of the form 3m + 1.
Rational primes that decompose in the field Q(sqrt(-3)). - N. J. A. Sloane, Dec 25 2017
Primes p dividing Sum_{k=0..p} binomial(2k, k) - 3 = A006134(p) - 3. - Benoit Cloitre, Feb 08 2003
Primes p such that tau(p) == 2 (mod 3) where tau(x) is the Ramanujan tau function (cf. A000594). - Benoit Cloitre, May 04 2003
Primes of the form x^2 + xy - 2y^2 = (x+2y)(x-y). - N. J. A. Sloane, May 31 2014
Primes of the form x^2 - xy + 7y^2 with x and y nonnegative. - T. D. Noe, May 07 2005
Primes p such that p^2 divides Sum_{m=1..2(p-1)} Sum_{k=1..m} (2k)!/(k!)^2. - Alexander Adamchuk, Jul 04 2006
A006512 larger than 5 (Greater of twin primes) is a subsequence of this. - Jonathan Vos Post, Sep 03 2006
A039701(A049084(a(n))) = A134323(A049084(a(n))) = 1. - Reinhard Zumkeller, Oct 21 2007
Also primes p such that the arithmetic mean of divisors of p^2 is an integer: sigma_1(p^2)/sigma_0(p^2) = C. (A000203(p^2)/A000005(p^2) = C). - Ctibor O. Zizka, Sep 15 2008
Fermat knew that these numbers can also be expressed as x^2 + 3y^2 and are therefore not prime in Z[omega], where omega is a complex cubic root of unity. - Alonso del Arte, Dec 07 2012
Primes of the form x^2 + xy + y^2 with x < y and nonnegative. Also see A007645 which also applies when x=y, adding an initial 3. - Richard R. Forberg, Apr 11 2016
For any term p in this sequence, let k = (p^2 - 1)/6; then A016921(k) = p^2. - Sergey Pavlov, Dec 16 2016; corrected Dec 18 2016
For the decomposition p=x^2+3*y^2, x(n) = A001479(n+1) and y(n) = A001480(n+1). - R. J. Mathar, Apr 16 2024

			Since 6 * 1 + 1 = 7 and 7 is prime, 7 is in the sequence. (Also 7 = 2^2 + 3 * 1^2 = (2 + sqrt(-3))(2 - sqrt(-3)).)
Since 6 * 2 + 1 = 13 and 13 is prime, 13 is in the sequence.
17 is prime but it is of the form 6m - 1 rather than 6m + 1, and is therefore not in the sequence.

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.
David A. Cox, Primes of the Form x^2 + ny^2. New York: Wiley (1989): 8.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 261.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
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].
C. Banderier, Calcul de (-3/p)
Barry Brent, Finite Field Models of Polynomials Interpolating Fourier Coefficients of Modular Functions for Hecke Groups, Integers (2024) Vol. 24, Art. No. A18. See p. 13.
F. S. Carey, On some cases of the Solutions of the Congruence z^p^(n-1)=1, mod p, Proceedings of the London Mathematical Society, Volume s1-33, Issue 1, November 1900, Pages 294-312.
A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
Jorma K. Merikoski, Pentti Haukkanen, and Timo Tossavainen, The congruence x^n = -a^n (mod m): Solvability and related OEIS sequences, Notes. Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 516-529. See p. 526.
K. G. Reuschle, Tafeln complexer Primzahlen, Königl. Akademie der Wissenschaften, Berlin, 1875, p. 1.
Neville Robbins, On the Infinitude of Primes of the Form 3k+1, Fib. Q., 43,1 (2005), 29-30.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Index to sequences related to decomposition of primes in quadratic fields

Cf. A045331, A242660.
For values of m see A024899. Primes of form 3n - 1 give A003627.
These are the primes arising in A024892, A024899, A034936.
A091178 gives prime index.
Cf. A006512, A007528.
Subsequence of A016921 and of A050931.
Cf. A004611 (multiplicative closure).

Filtered(List([0..110],k->6*k+1),n-> IsPrime(n)); # Muniru A Asiru, Mar 11 2019

a002476 n = a002476_list !! (n-1)
a002476_list = filter ((== 1) . (`mod` 6)) a000040_list
-- Reinhard Zumkeller, Jan 15 2013

(#~ 1&p:) >: 6 * i.1000 NB. Stephen Makdisi, May 01 2018

[n: n in [1..700 by 6] | IsPrime(n)]; // Vincenzo Librandi, Apr 05 2011

a := [ ]: for n from 1 to 400 do if isprime(6*n+1) then a := [ op(a), n ]; fi; od: A002476 := n->a[n];

Select[6*Range[100] + 1, PrimeQ[ # ] &] (* Stefan Steinerberger, Apr 06 2006 *)

select(p->p%3==1,primes(100)) \\ Charles R Greathouse IV, Oct 31 2012

From R. J. Mathar, Apr 03 2011: (Start)
Sum_{n >= 1} 1/a(n)^2 = A175644.
Sum_{n >= 1} 1/a(n)^3 = A175645. (End)
a(n) = 6*A024899(n) + 1. - Zak Seidov, Aug 31 2016
From Vaclav Kotesovec, May 02 2020: (Start)
Product_{k>=1} (1 - 1/a(k)^2) = 1/A175646.
Product_{k>=1} (1 + 1/a(k)^2) = A334481.
Product_{k>=1} (1 - 1/a(k)^3) = A334478.
Product_{k>=1} (1 + 1/a(k)^3) = A334477. (End)
Legendre symbol (-3, a(n)) = +1 and (-3, A007528(n)) = -1, for n >= 1. For prime 3 one sets (-3, 3) = 0. - Wolfdieter Lang, Mar 03 2021

A024899 Numbers k such that 6*k + 1 is prime.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 16, 17, 18, 21, 23, 25, 26, 27, 30, 32, 33, 35, 37, 38, 40, 45, 46, 47, 51, 52, 55, 56, 58, 61, 62, 63, 66, 68, 70, 72, 73, 76, 77, 81, 83, 87, 90, 91, 95, 96, 100, 101, 102, 103, 105, 107, 110, 112, 115, 118, 121, 122, 123, 125, 126, 128, 131, 135, 137

Offset: 1

Clark Kimberling

For all elements of this sequence there are no (x,y) positive integers such that a(n)=6*x*y+x+y or a(n)=6*x*y-x-y. - Pedro Caceres, Apr 19 2019

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

A002476 gives primes, A091178 gives prime index.

Complement of A046954 relative to A000027.

[n: n in [0..200]| IsPrime(6*n+1)] // Vincenzo Librandi, Nov 20 2010

a := [ ]: for n from 0 to 400 do if isprime(6*n+1) then a := [ op(a), n ]; fi; od: A002476 := n->a[n];

Select[Range@ 140, PrimeQ[6 # + 1] &] (* Michael De Vlieger, Jan 23 2018 *)

select(n->n%6==1,primes(100))\6 \\ Charles R Greathouse IV, Apr 28 2015

a(n) = A024892(n)/2 = (A034936(n)+1)/2. - Ray Chandler, Dec 26 2003

a(n) = (A002476(n)-1)/6. - Zak Seidov, Aug 31 2016

A034936 Numbers k such that 3*k + 4 is prime.

Original entry on oeis.org

1, 3, 5, 9, 11, 13, 19, 21, 23, 25, 31, 33, 35, 41, 45, 49, 51, 53, 59, 63, 65, 69, 73, 75, 79, 89, 91, 93, 101, 103, 109, 111, 115, 121, 123, 125, 131, 135, 139, 143, 145, 151, 153, 161, 165, 173, 179, 181, 189, 191, 199, 201, 203, 205, 209, 213, 219, 223, 229

Offset: 1

Jud McCranie

Related to hyperperfect numbers of a certain form.

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000
J. S. McCranie, A study of hyperperfect numbers, J. Int. Seqs. Vol. 3 (2000) #P00.1.3.

Cf. A038536 and A002476.
A002476 gives primes, A091178 gives prime index.
a(n) = A024892(n) - 1 = 2*A024899(n) - 1.
a(n) = A153183(n) - 2 = A107303(n) - 3.

[n: n in [1..1000] |IsPrime(3*n+4)] // Vincenzo Librandi, Nov 17 2010

lst={};Do[p=n+(n+1)+(n+3);If[PrimeQ[p],AppendTo[lst,n]],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, May 22 2009 *)
Select[Range[300],PrimeQ[3#+4]&] (* Harvey P. Dale, Aug 25 2016 *)

is(n)=isprime(3*n+4) \\ Charles R Greathouse IV, Jul 02 2013

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

A153183 Numbers k such that 3k-2 is prime.

Original entry on oeis.org

3, 5, 7, 11, 13, 15, 21, 23, 25, 27, 33, 35, 37, 43, 47, 51, 53, 55, 61, 65, 67, 71, 75, 77, 81, 91, 93, 95, 103, 105, 111, 113, 117, 123, 125, 127, 133, 137, 141, 145, 147, 153, 155, 163, 167, 175, 181, 183, 191, 193, 201, 203, 205, 207, 211, 215, 221, 225, 231

Offset: 1

Vincenzo Librandi, Dec 20 2008

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

Cf. A153184 (complement)

Cf. A088878 (3p-2 is prime), A024892 (3*k+1 is prime).

[ n: n in [1..235] | IsPrime(3*n-2) ]; // Klaus Brockhaus, Dec 21 2008

Select[Range[1, 250], PrimeQ[(3*#)-2]&] (* Vincenzo Librandi, Sep 24 2012 *)

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

a(n) = A024892(n)+1. - Klaus Brockhaus, Dec 21 2008

More terms from Klaus Brockhaus, Dec 21 2008

A070847 Smallest prime == 1 mod (3n).

Original entry on oeis.org

7, 7, 19, 13, 31, 19, 43, 73, 109, 31, 67, 37, 79, 43, 181, 97, 103, 109, 229, 61, 127, 67, 139, 73, 151, 79, 163, 337, 349, 181, 373, 97, 199, 103, 211, 109, 223, 229, 937, 241, 739, 127, 1033, 397, 271, 139, 283, 433, 883, 151, 307, 157, 3181, 163, 331, 337

Offset: 1

Amarnath Murthy, May 15 2002

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

Cf. A070846, A070848, A070849, A070850, A070851, A070852, A070853.
Cf. A034694.
Cf. A024892 (n such that a(n)=3*n+1).
Cf. A002476.

f:= proc(n) local k,d;
  if n::even then d:= 3*n else d:= 6*n fi;
  for k from 1 by d do if isprime(k) then return k fi od
end proc:
map(f, [$1..100]); # Robert Israel, Sep 19 2019

a[n_] := Module[{k, d}, If[EvenQ[n], d = 3n, d = 6n]; For[k = 1, True, k += d, If[PrimeQ[k], Return[k]]]];
Array[a, 100] (* Jean-François Alcover, Jun 11 2020, after Maple *)

for(n=1,80,s=1; while((isprime(s)*s-1)%(3*n)>0,s++); print1(s,","))

More terms from Benoit Cloitre, May 18 2002

A130800 Numbers k such that both 2k+1 and 3k+1 are primes.

Original entry on oeis.org

2, 6, 14, 20, 26, 36, 50, 54, 74, 90, 116, 140, 146, 174, 200, 204, 210, 224, 230, 270, 284, 306, 330, 336, 350, 354, 384, 404, 410, 426, 440, 476, 510, 516, 554, 564, 596, 600, 624, 644, 650, 704, 714, 726, 740, 746, 834, 846, 894, 930, 944, 950, 1026, 1040

Offset: 1

Max Alekseyev, Jul 18 2007

nonn

Also: k such that A033570(k) is semiprime. All terms are congruent to 0 or 2 modulo 6. - M. F. Hasler, Dec 13 2019

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

Intersection of A005097 and A024892. - M. F. Hasler, Dec 13 2019

Cf. A033570; A255584: semiprimes of the form (4*n+1)*(6*n+1).

[n: n in [0..500] | IsPrime(2*n+1) and IsPrime(3*n+1)]; // Vincenzo Librandi, Nov 23 2010

Select[Range[1100],AllTrue[{2,3}#+1,PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 17 2016 *)

select( is_A130800(n)=isprime(2*n+1)&&isprime(3*n+1), [1..1111]) \\ M. F. Hasler, Dec 13 2019

a(n) = 2*A255607(n). - M. F. Hasler, Dec 13 2019

More terms from Vincenzo Librandi, Mar 26 2010

A112772 Semiprimes of the form 6n+2.

Original entry on oeis.org

14, 26, 38, 62, 74, 86, 122, 134, 146, 158, 194, 206, 218, 254, 278, 302, 314, 326, 362, 386, 398, 422, 446, 458, 482, 542, 554, 566, 614, 626, 662, 674, 698, 734, 746, 758, 794, 818, 842, 866, 878, 914, 926, 974, 998, 1046, 1082, 1094, 1142, 1154, 1202, 1214

Offset: 1

Jonathan Vos Post and Ray Chandler, Oct 15 2005

{6} + A112772 + A112774 = A100484 = 2*A000040.

Rado showed that for a given Bernoulli number B_n there exist infinitely many Bernoulli numbers B_m having the same denominator. As a special case, if n = 2p where p is an odd prime p == 1 (mod 3), then the denominator of the Bernoulli number B_n equals 6. - Bernd C. Kellner, Mar 21 2018

Harvey P. Dale, Table of n, a(n) for n = 1..1000
R. Rado, A note on the Bernoullian numbers, J. London Math. Soc. 9 (1934) 88-90.

Subsequence of A051222. - Bernd C. Kellner, Mar 21 2018

Cf. A027642.

IsSemiprime:= func; [s: n in [0..210] | IsSemiprime(s) where s is 6*n + 2]; // Vincenzo Librandi, Sep 22 2012

Select[6Range[0,300]+2,PrimeOmega[#]==2&] (* Harvey P. Dale, Oct 04 2011 *)

2*select(n->n%3==1,primes(100)) \\ Charles R Greathouse IV, Sep 22 2012

a(n) = 2 * A002476(n) = 6 * A024892(n) + 2.

denominator(Bernoulli(a(n))) = 6. - Bernd C. Kellner, Mar 21 2018

A089953 Numbers n such that 3*n+7 is prime.

Original entry on oeis.org

0, 2, 4, 8, 10, 12, 18, 20, 22, 24, 30, 32, 34, 40, 44, 48, 50, 52, 58, 62, 64, 68, 72, 74, 78, 88, 90, 92, 100, 102, 108, 110, 114, 120, 122, 124, 130, 134, 138, 142, 144, 150, 152, 160, 164, 172, 178, 180, 188, 190, 198, 200, 202, 204, 208, 212, 218, 222, 228

Offset: 1

Giovanni Teofilatto, Jan 12 2004

nonn

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.

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

Cf. A002476 gives primes, A034936, A024892, A024899.

is(n)=isprime(3*n+7) \\ Charles R Greathouse IV, May 22 2017

a(n) = A034936(n)-1 = A024892(n)-2 = 2*A024899(n)-2.

Offset corrected by Arkadiusz Wesolowski, Aug 09 2011

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

cp's OEIS Frontend

A002476 Primes of the form 6m + 1.

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A024899 Numbers k such that 6*k + 1 is prime.

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A034936 Numbers k such that 3*k + 4 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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A153183 Numbers k such that 3k-2 is prime.

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A070847 Smallest prime == 1 mod (3n).

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Maple

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A130800 Numbers k such that both 2k+1 and 3k+1 are primes.

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