A024899 -id:A024899 - 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

A002822 Numbers m such that 6m-1, 6m+1 are twin primes.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 12, 17, 18, 23, 25, 30, 32, 33, 38, 40, 45, 47, 52, 58, 70, 72, 77, 87, 95, 100, 103, 107, 110, 135, 137, 138, 143, 147, 170, 172, 175, 177, 182, 192, 205, 213, 215, 217, 220, 238, 242, 247, 248, 268, 270, 278, 283, 287, 298, 312, 313, 322, 325

Offset: 1

N. J. A. Sloane

6m-1 and 6m+1 are twin primes iff m is not of the form 6ab +- a +- b. - Jon Perry, Feb 01 2002
The above equivalence was rediscovered by Balestrieri, see link. - Charles R Greathouse IV, Jul 05 2011
Even terms correspond to twin primes of the form (4k - 1, 4k + 1), odd terms to twin primes of the form (4k + 1, 4k + 3). - Lekraj Beedassy, Apr 03 2002
A002822 U A067611 U A171696 = A001477. - Juri-Stepan Gerasimov, Feb 14 2010
From Bob Selcoe, Nov 28 2014: (Start)
Except for a(1)=1, all numbers in this sequence are congruent to (0, 2 or 3) mod 5.
It appears that when a(n)=6j, then j is also in the sequence (e.g., 138 = 6*23; 312 = 6*52). This also appears to hold for sequence A191626. If true, then it suggests that when seeking large twin primes, good candidates might be 36*a(n) +- 1, n >= 2.
Conjecture: There is at least one number in the sequence in the interval [5k, 7k] inclusive, k >= 1. If true, then the twin prime conjecture also is true.
(End)
A counterexample to "It appears that ...": Take j = 63. Then 6j = 378 and 36j = 2268. Now 379, 2267, and 2269 are prime, but 377 = 13 * 29. The sequence of counterexamples is A263282. - Jason Kimberley, Oct 13 2015
Dinculescu calls all terms in the sequence "twin ranks", and all other positive integers "non-ranks", see links. Non-ranks are given by the formula kp +- round(p/6) for positive integers k and primes p > 4, while twin ranks (this sequence) cannot be represented as kp +- round(p/6) for any k, p > 4. Here round(p/6) is the nearest integer to p/6. - Alexei Kourbatov, Jan 03 2015
Number of terms less than 10^k: 0, 5, 25, 142, 810, 5330, 37915, ... - Muniru A Asiru, Jan 24 2018
6m-1 and 6m+1 are twin primes iff 36m^2-1 is semiprime. It is algebraically provable that 36m^2-1 having any factor of the form 6k+-1 is equivalent to the statement that m is congruent to +-k (mod (6k+-1)). Other than the trivial case m=k, the fact of such a congruence means 36m^2-1 has a factor other than 6m-1 and 6m+1, and is not semiprime. Thus, {a(n)} lists the numbers m such that for all k < m, m is not congruent to +-k modulo (6k+-1). This is an alternative formulation of the results of Dinculescu referenced above. - Keith Backman, Apr 25 2021
Other than a(1)=1, it is provable that a(n) is not a square unless it is a multiple of 5, and a(n) is not a cube unless it is a multiple of 7. Examples of the former include a(11)=5^2=25, a(26)=10^2=100, and a(166)=35^2=1225; examples of the latter are rarer, including a(1531)=28^3=21952 and a(4163)=42^3=74088. - Keith Backman, Jun 26 2021

W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, p. 69.
W. Sierpiński, A Selection of Problems in the Theory of Numbers. Macmillan, NY, 1964, p. 120.
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).

T. D. Noe, Table of n, a(n) for n = 1..10000
F. Balestrieri, An Equivalent Problem To The Twin Prime Conjecture, arXiv:1106.6050v1 [math.GM], 2011.
A. Dinculescu, On Some Infinite Series Related to the Twin Primes, The Open Mathematics Journal, 5 (2012), 8-14.
A. Dinculescu, The Twin Primes Seen from a Different Perspective, The British Journal of Mathematics & Computer Science, 3 (2013), Issue 4, 691-698.
A. Dinculescu, On the Numbers that Determine the Distribution of Twin Primes, Surveys in Mathematics and its Applications, 13 (2018), 171-181.
S. W. Golomb, Problem E969, Solution, Amer. Math. Monthly, 58 (1951), 338; 59 (1952), 44.
S. W. Golomb, Letter to N. J. A. Sloane, Mar 26 1984
Matthew A. Myers, Comments on A002822, Letter to N. J. A. Sloane, Dec 04 2018

Complement of A067611.
Intersection of A024898 and A024899.
A191626 is a subsequence.
Cf. A014574, A263282.

a002822 n = a002822_list !! (n-1)
a002822_list = f a000040_list where
   f (q:ps'@(p:ps)) | p > q + 2 || r > 0 = f ps'
                    | otherwise = y : f ps where (y,r) = divMod (q + 1) 6
-- Reinhard Zumkeller, Jul 13 2014

[n: n in [1..200] | IsPrime(6*n+1) and IsPrime(6*n-1)] // Vincenzo Librandi, Nov 21 2010

select(n -> isprime(6*n-1) and isprime(6*n+1), [$1..1000]); # Robert Israel, Jan 11 2015

Select[ Range[350], PrimeQ[6# - 1] && PrimeQ[6# + 1] & ]
Select[Range[400],AllTrue[6#+{1,-1},PrimeQ]&] (* Harvey P. Dale, Jul 27 2022 *)
#/6&/@Select[Range[6,2500,6],AllTrue[#+{1,-1},PrimeQ]&] (* Harvey P. Dale, Mar 31 2023 *)

select(primes(100),n->isprime(n-2)&&n>5)\6 \\ Charles R Greathouse IV, Jul 05 2011

p=5; forprime(q=5, 1e4, if(q-p==2, print1((p+1)/6", ")); p=q); \\ Altug Alkan, Oct 13 2015

list(lim)=my(v=List(),p=5); forprime(q=7,6*lim+1, if(q-p==2, listput(v,q\6)); p=q); Vec(v) \\ Charles R Greathouse IV, Dec 03 2016

a(n) = A014574(n+1)/6. - Ivan N. Ianakiev, Aug 19 2013

More terms from Larry Reeves (larryr(AT)acm.org), Mar 27 2001

A007693 Primes p such that 6*p + 1 is also prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 23, 37, 47, 61, 73, 83, 101, 103, 107, 131, 137, 151, 173, 181, 233, 241, 257, 263, 271, 277, 283, 293, 311, 313, 331, 347, 367, 373, 397, 443, 461, 467, 503, 557, 577, 593, 601, 607, 641, 653, 661, 683, 727, 751, 761, 773, 787, 797, 853

Offset: 1

N. J. A. Sloane and Robert G. Wilson v

Joe Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 83.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
Andrew Granville, Sophie Germain's theorem for prime pairs p, 6p+1, J. Number Theory 27 (1987), no. 1, 63-72.

Cf. A002476, A016921, A024899, A051644, A091178, A023256 (subset: 6*(6p+1)+1 also prime).

Prime terms of A024899.

[n: n in [0..1000] | IsPrime(n) and IsPrime(6*n+1)]; // Vincenzo Librandi, Nov 18 2010

Select[Prime@Range[150], PrimeQ[6# + 1] &] (* Ray Chandler, Mar 14 2007  *)

isok(k) = isprime(k) && isprime(6*k+1); \\ Amiram Eldar, Feb 24 2025

a(n) = (A051644(n)-1)/6.

Extended by Ray Chandler, Mar 14 2007

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

A024892 Numbers k such that 3*k+1 is prime.

Original entry on oeis.org

2, 4, 6, 10, 12, 14, 20, 22, 24, 26, 32, 34, 36, 42, 46, 50, 52, 54, 60, 64, 66, 70, 74, 76, 80, 90, 92, 94, 102, 104, 110, 112, 116, 122, 124, 126, 132, 136, 140, 144, 146, 152, 154, 162, 166, 174, 180, 182, 190, 192, 200, 202, 204, 206, 210, 214, 220, 224, 230, 236, 242, 244, 246

Offset: 1

Clark Kimberling

Every prime (with the exception of 3) can be expressed as 3*k+1 or 3*k-1. - César Aguilera, Apr 13 2013
The associated prime A002476(n) has a unique representation as x^2 + x*y - 2*y^2 = (x + 2*y)*(x-y) with positive integers, namely (x(n), y(n)) = (a(n) + 1, a(n)). See the N. J. A. Sloane, May 31 2014, comment on A002476. - Wolfdieter Lang, Feb 09 2016
For all elements of this sequence there are no (x,y) positive integers such that a(n) = 3*x*y + x + y or a(n) = 3*x*y - x - y. - Pedro Caceres, Jan 28 2021

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

Cf. A002476 (associated primes), A091178 (gives prime index).

Cf. A153183, A107303.

[n: n in [1..1000] | IsPrime(3*n+1)]; // Vincenzo Librandi, Nov 20 2010

Select[Range[250], PrimeQ[3# + 1] &] (* Vincenzo Librandi, Sep 25 2012 *)

is(n)=isprime(3*n+1) \\ Charles R Greathouse IV, Feb 17 2017

a(n) = (A002476(n) - 1)/3. See the name.
a(n) = 2*A024899(n) = A034936(n) + 1.
a(n) = A153183(n) - 1 = A107303(n) - 2.

A056956 Numbers n such that 6n+1 and 6n+5 are both primes.

Original entry on oeis.org

1, 2, 3, 6, 7, 11, 13, 16, 17, 18, 21, 27, 32, 37, 38, 46, 51, 52, 58, 63, 66, 73, 76, 77, 81, 83, 102, 107, 112, 123, 126, 128, 137, 142, 143, 146, 147, 151, 156, 161, 168, 181, 182, 202, 213, 216, 217, 237, 238, 241, 247, 248, 258, 261, 263, 266, 268, 277, 282

Offset: 1

Henry Bottomley, Jul 18 2000

nonn

Note that if prime p>3 then p mod 6 = 1 or 5.

			a(2)=2 since 6*2+1=13 and 6*2+5=17 are both prime.

M. F. Hasler, Table of n, a(n) for n = 1..5000

Cf. A002822, A024898, A024899, A059325.

Select[Range[300], And @@ PrimeQ /@ ({1, 5} + 6#) &] (* Ray Chandler, Jun 29 2008 *)

is(n)=isprime(n*6+1)&&isprime(n*6+5) \\ M. F. Hasler, Apr 05 2017

a(n) = (A023200(n+1)-1)/6 = (A046132(n+1)-5)/6 = A047847(n+1)/3

a(n) = floor(A087679(n+1)/6). - M. F. Hasler, Apr 05 2017

Edited by N. J. A. Sloane, Nov 07 2006

A046954 Numbers k such that 6*k + 1 is nonprime.

Original entry on oeis.org

0, 4, 8, 9, 14, 15, 19, 20, 22, 24, 28, 29, 31, 34, 36, 39, 41, 42, 43, 44, 48, 49, 50, 53, 54, 57, 59, 60, 64, 65, 67, 69, 71, 74, 75, 78, 79, 80, 82, 84, 85, 86, 88, 89, 92, 93, 94, 97, 98, 99, 104, 106, 108, 109, 111, 113, 114, 116, 117, 119, 120, 124, 127, 129, 130, 132, 133, 134, 136, 139, 140

Offset: 1

Felice Russo

nonn

Equals A171696 U A121763; A121765 U A171696 = A046953; A121763 U A121765 = A067611 where A067611 U A002822 U A171696 = A001477. - Juri-Stepan Gerasimov, Feb 13 2010, Feb 15 2010

These numbers (except 0) can be written as 6xy +-(x+y) for x > 0, y > 0. - Ron R Spencer, Aug 01 2016

			a(2)=8 because 6*8 + 1 = 49, which is composite.

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

Cf. A047845 (2n+1), A045751 (4n+1), A127260 (8n+1).

Cf. A046953, A008588, A016921, subsequence of A067611, complement of A024899.

Filtered([0..250], k-> not IsPrime(6*k+1)) # G. C. Greubel, Feb 21 2019

a046954 n = a046954_list !! (n-1)
a046954_list = map (`div` 6) $ filter ((== 0) . a010051' . (+ 1)) [0,6..]
-- Reinhard Zumkeller, Jul 13 2014

[n: n in [0..250] | not IsPrime(6*n+1)]; // G. C. Greubel, Feb 21 2019

remove(k-> isprime(6*k+1), [$0..140])[]; # Muniru A Asiru, Feb 22 2019

a = Flatten[Table[If[PrimeQ[6*n + 1] == False, n, {}], {n, 0, 50}]] (* Roger L. Bagula, May 17 2007 *)
Select[Range[0, 200], !PrimeQ[6 # + 1] &] (* Vincenzo Librandi, Sep 27 2013 *)

is(n)=!isprime(6*n+1) \\ Charles R Greathouse IV, Aug 01 2016

[n for n in (0..250) if not is_prime(6*n+1)] # G. C. Greubel, Feb 21 2019

Edited by N. J. A. Sloane, Aug 08 2008 at the suggestion of R. J. Mathar
Corrected by Juri-Stepan Gerasimov, Feb 13 2010, Feb 15 2010
Corrected by Vincenzo Librandi, Sep 27 2013

A051644 Primes of the form 6*p + 1 where p is also prime.

Original entry on oeis.org

13, 19, 31, 43, 67, 79, 103, 139, 223, 283, 367, 439, 499, 607, 619, 643, 787, 823, 907, 1039, 1087, 1399, 1447, 1543, 1579, 1627, 1663, 1699, 1759, 1867, 1879, 1987, 2083, 2203, 2239, 2383, 2659, 2767, 2803, 3019, 3343, 3463, 3559, 3607, 3643, 3847, 3919

Offset: 1

Labos Elemer

Analogous to A005385; can be called 6-safe primes.

			103 is in the sequence because both 17 and 6*17 + 1 = 103 are primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)

Cf. A005384, A005385, A002476, A007693, A016921, A024899, A091178.

Select[1 + 6Prime@Range[120], PrimeQ] (* Ray Chandler, Mar 14 2007 *)

isok(k) = isprime(k) && k % 6 == 1 && isprime((k-1)/6); \\ Amiram Eldar, Feb 24 2025

a(n) = 6*A007693(n) + 1.

Edited, corrected and extended by Ray Chandler, Mar 14 2007

cp's OEIS Frontend

A139394 a(n) = A024899(n)-A024898(n).

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A002476 Primes of the form 6m + 1.

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A002822 Numbers m such that 6m-1, 6m+1 are twin primes.

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A007693 Primes p such that 6*p + 1 is also prime.

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Magma

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

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Magma

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

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