A034936 -id:A034936 - cp's OEIS frontend

A173177 Numbers k such that 2k+3 is a prime of the form 3*A034936(m) + 4.

2, 5, 8, 14, 17, 20, 29, 32, 35, 38, 47, 50, 53, 62, 68, 74, 77, 80, 89, 95, 98, 104, 110, 113, 119, 134, 137, 140, 152, 155, 164, 167, 173, 182, 185, 188, 197, 203, 209, 215, 218, 227, 230, 242, 248, 260, 269, 272, 284, 287, 299

Offset: 1

Eric Desbiaux, Feb 11 2010

With Bachet-Bézout theorem implicating Gauss Lemma and the Fundamental Theorem of Arithmetic,
for k > 1, k = 2*a + 3*b (a and b integers)
first type
A001477 = (2*A080425) + (3*A008611)
A000040 = (2*A039701) + (3*A157966)
A024893 Numbers k such that 3*k + 2 is prime
A034936 Numbers k such that 3*k + 4 is prime
OR
second type
A001477 = (2*A028242) + (3*A059841)
A000040 = (2*A067076) + (3*1)
A067076 Numbers k such that 2*k + 3 is prime
   k   a b OR a b
  --   - -    - -
   0   0 0    0 0
   1   - -    - -
   2   1 0    1 0
   3   0 1    0 1
   4   2 0    2 0
   5   1 1    1 1
   6   0 2    3 0
   7   2 1    2 1
   8   1 2    4 0
   9   0 3    3 1
  10   2 2    5 0
  11   1 3    4 1
  12   0 4    6 0
  13   2 3    5 1
  14   1 4    7 0
  15   0 5    6 1
  ...
  2* 2 + 3 OR 3* 1 + 4 =  7;
  2* 5 + 3 OR 3* 3 + 4 = 13;
  2* 8 + 3 OR 3* 5 + 4 = 19;
  2*14 + 3 OR 3* 9 + 4 = 31;
  2*17 + 3 OR 3*11 + 4 = 37;
  2*20 + 3 OR 3*13 + 4 = 43;
  2*29 + 3 OR 3*19 + 4 = 61;
  2*32 + 3 OR 3*21 + 4 = 67;
  2*35 + 3 OR 3*23 + 4 = 73.
A034936 Numbers k such that 3k+4 is prime.
A002476 Primes of the form 6k+1.
A024899 Nonnegative integers k such that 6k+1 is prime.
2, 5, 8, 14, 17, 20, ... = (3*(4*A024899 - A034936) - 5)/2.

Chris K. Caldwell, FAQ: Are all primes (past 2 and 3) of the forms 6n+1 and 6n-1?

Cf. A067076, A034936, A002476, A024899.

Select[Range[300],PrimeQ[2#+3]&&Divisible[2#-1,3]&] (* Harvey P. Dale, Aug 25 2016 *)

More terms from Harvey P. Dale, Aug 25 2016

A002476 Primes of the form 6m + 1.

Original entry on oeis.org

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

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.

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

A023278 Primes that remain prime through 3 iterations of function f(x) = 3x + 4.

Original entry on oeis.org

23, 683, 1663, 2753, 3203, 6073, 6323, 7523, 8243, 9293, 9613, 15173, 19913, 21023, 21683, 25183, 26633, 29663, 34613, 34703, 39293, 41953, 43283, 45533, 52813, 59393, 62473, 65053, 66763, 71713, 71993, 81533, 81953, 84523, 87833, 103843, 104183

Offset: 1

David W. Wilson

nonn

Primes p such that 3*p+4, 9*p+16 and 27*p+52 are also primes. - Vincenzo Librandi, Aug 04 2010

John Cerkan, Table of n, a(n) for n = 1..10000

Subsequence of A023209, A023247, and of A034936.

[n: n in [1..150000] | IsPrime(n) and IsPrime(3*n+4) and IsPrime(9*n+16) and IsPrime(27*n+52)] // Vincenzo Librandi, Aug 04 2010

Select[Prime@ Range[10^4], Times @@ Boole@ PrimeQ@ Rest@ NestList[3 # + 4 &, #, 3] > 0 &] (* Michael De Vlieger, Sep 19 2016 *)

is(n)=isprime(n) && isprime(3*n+4) && isprime(9*n+16) && isprime(27*n+52) \\ Charles R Greathouse IV, Sep 20 2016

a(n) == 3 (mod 10). - John Cerkan, Sep 16 2016

A023308 Primes that remain prime through 4 iterations of the function f(x) = 3x + 4.

Original entry on oeis.org

3203, 21683, 34613, 52813, 103843, 116933, 117443, 165443, 172933, 193603, 195053, 213973, 226783, 321053, 322193, 357613, 360323, 362233, 363403, 368743, 472393, 474143, 496333, 518543, 528673, 569083, 571303, 631853, 654623, 714893, 758503

Offset: 1

David W. Wilson

nonn

Primes p such that 3*p+4, 9*p+16, 27*p+52 and 81*p+160 are also primes. - Vincenzo Librandi, Aug 04 2010

All a(n) == 33 or 53 (mod 70). - John Cerkan, Oct 04 2016

John Cerkan, Table of n, a(n) for n = 1..10000

Subsequence of A023209, A023247, A023278, and A034936.

Filtered([1..760000],n->IsPrime(n) and IsPrime(3*n+4) and IsPrime(9*n+16) and IsPrime(27*n+52) and IsPrime(81*n+160)); # Muniru A Asiru, Dec 07 2018

[n: n in [1..1000000] | IsPrime(n) and IsPrime(3*n+4) and IsPrime(9*n+16) and IsPrime(27*n+52) and IsPrime(81*n+160)] // Vincenzo Librandi, Aug 04 2010

select(n->isprime(n) and isprime(3*n+4) and isprime(9*n+16) and isprime(27*n+52) and isprime(81*n+160),[$1..760000]); # Muniru A Asiru, Dec 07 2018

Select[Prime[Range[10000]], Union[PrimeQ[NestList[(3# + 4 &), #, 4]]] == {True} &] (* Alonso del Arte, Nov 30 2018 *)

is(n) = my(x=3*n+4, i=0); while(1, if(!ispseudoprime(x), return(0), i++); if(i==4, return(1)); x=3*x+4)
forprime(p=1, 760000, if(is(p), print1(p, ", "))) \\ Felix Fröhlich, Dec 07 2018

A106068 Primes p such that 3p + 4 and 4p + 3 are primes.

Original entry on oeis.org

5, 11, 19, 31, 41, 59, 89, 109, 151, 179, 181, 229, 241, 331, 349, 389, 439, 509, 521, 599, 661, 719, 769, 839, 881, 929, 1019, 1039, 1129, 1229, 1291, 1409, 1451, 1481, 1549, 1669, 1741, 1759, 1801, 1811, 2111, 2131, 2539, 2621, 2671, 2699, 2819, 2879

Offset: 1

Zak Seidov, May 07 2005

nonn

Prime terms in A124855.

Marius A. Burtea, Table of n, a(n) for n = 1..534

Cf. A034936, A095278, A124855.

[p: p in PrimesUpTo(5000)|IsPrime(3*p+4) and IsPrime(4*p+3)] // Vincenzo Librandi, Jan 30 2011

Select[Prime[Range[450]], PrimeQ[4#+3]&&PrimeQ[3#+4]&]

isok(p) = isprime(p) && isprime(3*p+4) && isprime(4*p+3); \\ Michel Marcus, Oct 12 2018

Extended by Ray Chandler, Mar 14 2007

A153282 Numbers k such that 3*k + 4 is not prime.

Original entry on oeis.org

0, 2, 4, 6, 7, 8, 10, 12, 14, 15, 16, 17, 18, 20, 22, 24, 26, 27, 28, 29, 30, 32, 34, 36, 37, 38, 39, 40, 42, 43, 44, 46, 47, 48, 50, 52, 54, 55, 56, 57, 58, 60, 61, 62, 64, 66, 67, 68, 70, 71, 72, 74, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88, 90, 92, 94, 95, 96, 97, 98, 99, 100

Offset: 1

Vincenzo Librandi, Dec 23 2008

Complement of A034936. - Omar E. Pol, Jan 18 2009

			Distribution of the odd terms in the following triangular array:
*;
*,7;
*,*,15;
*,*,*,*;
*,17.*,*,39;
*,*,29,*,*,55;
*,*, *,*,*,*, *;
*,27,*,*,61,*,*,95;
*,*,43,*,*,81,*,*,119;
*,*, *,*,*,*, *,*, *, *;
*,37,*,*,83,*,*,129,*,*,175;
*,*,57,*,*,107,*,*,157,*,*,207; etc.
where * marks the non-integer values of (4*h*k + 2*k + 2*h - 3)/3 with h >= k >= 1. - _Vincenzo Librandi_, Jan 17 2013

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

Cf. A034936.

[n: n in [0..150] | not IsPrime(3*n + 4)]; // Vincenzo Librandi, Jan 12 2013

Select[Range[0, 200], !PrimeQ[3*# + 4]&] (* Vincenzo Librandi, Jan 12 2013 *)

Edited (and terms confirmed) by N. J. A. Sloane, Jan 18 2009

cp's OEIS Frontend

A173177 Numbers k such that 2k+3 is a prime of the form 3*A034936(m) + 4.

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A002476 Primes of the form 6m + 1.

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Magma

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

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Magma

Maple

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

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Magma

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

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PARI

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

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Magma

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PARI

A023278 Primes that remain prime through 3 iterations of function f(x) = 3x + 4.

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