A040117 -id:A040117 - cp's OEIS frontend

A106856 Primes of the form x^2 + xy + 2y^2, with x and y nonnegative.

2, 11, 23, 37, 43, 53, 71, 79, 107, 109, 127, 137, 149, 151, 163, 193, 197, 211, 233, 239, 263, 281, 317, 331, 337, 373, 389, 401, 421, 431, 443, 463, 487, 491, 499, 541, 547, 557, 569, 599, 613, 617, 641, 653, 659, 673, 683, 739, 743, 751, 757, 809, 821

Offset: 1

T. D. Noe, May 09 2005, Apr 28 2008

Discriminant=-7. Binary quadratic forms ax^2 + bxy + cy^2 have discriminant d = b^2 - 4ac.

Consider sequences of primes produced by forms with -100

The Mathematica function QuadPrimes2 is useful for finding the primes less than "lim" represented by the positive definite quadratic form ax^2 + bxy + cy^2 for any a, b and c satisfying a>0, c>0, and discriminant d<0. It does this by examining all x>=0 and y>=0 in the ellipse ax^2 + bxy + cy^2 <= lim. To find the primes generated by positive and negative x and y, compute the union of QuadPrimes2[a,b,c,lim] and QuadPrimes2[a,-b,c,lim]. - T. D. Noe, Sep 01 2009

For other programs see the "Binary Quadratic Forms and OEIS" link.

David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.
L. E. Dickson, History of the Theory of Numbers, Vol. 3, Chelsea, 1923.

Zak Seidov and N. J. A. Sloane, Table of n, a(n) for n = 1..10000 (The first 1225 terms were found by Zak Seidov)
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Discriminants in the range -3 to -100: A007645 (d=-3), A002313 (d=-4), A045373, A106856 (d=-7), A033203 (d=-8), A056874, A106857 (d=-11), A002476 (d=-12), A033212, A106858-A106861 (d=-15), A002144, A002313 (d=-16), A106862-A106863 (d=-19), A033205, A106864-A106865 (d=-20), A106866-A106869 (d=-23), A033199, A084865 (d=-24), A002476, A106870 (d=-27), A033207 (d=-28), A033221, A106871-A106874 (d=-31), A007519, A007520, A106875-A106876 (d=-32), A106877-A106881 (d=-35), A040117, A068228, A106882 (d=-36), A033227, A106883-A106888 (d=-39), A033201, A106889 (d=-40), A106890-A106891 (d=-43), A033209, A106282, A106892-A106893 (d=-44), A033232, A106894-A106900 (d=-47), A068229 (d=-48), A106901-A106904 (d=-51), A033210, A106905-A106906 (d=-52), A033235, A106907-A106913 (d=-55), A033211, A106914-A106917 (d=-56), A106918-A106922 (d=-59), A033212, A106859 (d=-60), A106923-A106930 (d=-63), A007521, A106931 (d=-64), A106932-A106933 (d=-67), A033213, A106934-A106938 (d=-68), A033246, A106939-A106948 (d=-71), A106949-A106950 (d=-72), A033212, A106951-A106952 (d=-75), A033214, A106953-A106955 (d=-76), A033251, A106956-A106962 (d=-79), A047650, A106963-A106965 (d=-80), A106966-A106970 (d=-83), A033215, A102271, A102273, A106971-A106974 (d=-84), A033256, A106975-A106983 (d=-87), A033216, A106984 (d=-88), A106985-A106989 (d=-91), A033217 (d=-92), A033206, A106990-A107001 (d=-95), A107002-A107008 (d=-96), A107009-A107013 (d=-99).
Other collections of quadratic forms: A139643, A139827.
For a more comprehensive list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
Cf. also A242660.

QuadPrimes2[a_, b_, c_, lmt_] := Module[{p, d, lst = {}, xMax, yMax}, d = b^2 - 4a*c; If[a > 0 && c > 0 && d < 0, xMax = Sqrt[lmt/a]*(1+Abs[b]/Floor[Sqrt[-d]])]; Do[ If[ 4c*lmt + d*x^2 >= 0, yMax = ((-b)*x + Sqrt[4c*lmt + d*x^2])/(2c), yMax = 0 ]; Do[p = a*x^2 + b*x*y + c*y^2; If[ PrimeQ[ p]  && p <= lmt && !MemberQ[ lst, p], AppendTo[ lst, p]], {y, 0, yMax}], {x, 0, xMax}]; Sort[ lst]];
QuadPrimes2[1, 1, 2, 1000]
(This is a corrected version of the old, incorrect, program QuadPrimes. - N. J. A. Sloane, Jun 15 2014)
max = 1000; Table[yy = {y, 1, Floor[Sqrt[8 max - 7 x^2]/4 - x/4]}; Table[ x^2 + x y + 2 y^2, yy // Evaluate], {x, 0, Floor[Sqrt[max]]}] // Flatten // Union // Select[#, PrimeQ]& (* Jean-François Alcover, Oct 04 2018 *)

list(lim)=my(q=Qfb(1,1,2), v=List([2])); forprime(p=2, lim, if(vecmin(qfbsolve(q, p))>0, listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Aug 05 2016

Removed old Mathematica programs - T. D. Noe, Sep 09 2009

Edited (pointed out error in QuadPrimes, added new version of program, checked and extended b-file). - N. J. A. Sloane, Jun 06 2014

A007528 Primes of the form 6k-1.

Original entry on oeis.org

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

For values of k see A024898.
Also primes p such that p^q - 2 is not prime where q is an odd prime. These numbers cannot be prime because the binomial p^q = (6k-1)^q expands to 6h-1 some h. Then p^q-2 = 6h-1-2 is divisible by 3 thus not prime. - Cino Hilliard, Nov 12 2008
a(n) = A211890(3,n-1) for n <= 4. - Reinhard Zumkeller, Jul 13 2012
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
From Bernard Schott, Feb 14 2019: (Start)
A theorem due to Andrzej Mąkowski: every integer greater than 161 is the sum of distinct primes of the form 6k-1. Examples: 162 = 5 + 11 + 17 + 23 + 47 + 59; 163 = 17 + 23 + 29 + 41 + 53. (See Sierpiński and David Wells.)
{2,3} Union A002476 Union {this sequence} = A000040.
Except for 2 and 3, all Sophie Germain primes are of the form 6k-1.
Except for 3, all the lesser of twin primes are also of the form 6k-1.
Dirichlet's theorem on arithmetic progressions states that this sequence is infinite. (End)
For all elements of this sequence p=6*k-1, there are no (x,y) positive integers such that k=6*x*y-x+y. - Pedro Caceres, Apr 06 2019

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.
A. Mąkowski, Partitions into unequal primes, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 8 (1960), 125-126.
Wacław Sierpiński, Elementary Theory of Numbers, p. 144, Warsaw, 1964.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, Revised edition, 1997, p. 127.

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].
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.
Amelia Carolina Sparavigna, The Pentagonal Numbers and their Link to an Integer Sequence which contains the Primes of Form 6n-1, Politecnico di Torino (Italy, 2021).
Amelia Carolina Sparavigna, Binary operations inspired by generalized entropies applied to figurate numbers, Politecnico di Torino (Italy, 2021).
Wikipedia, Dirichlet's theorem on arithmetic progressions.

Intersection of A016969 and A000040.
Cf. A003627, A010051, A117047, A132231, A214360, A057145.
Prime sequences A# (k,r) of the form k*n+r with 0 <= r <= k-1 (i.e., primes == r (mod k), or primes p with p mod k = r) and gcd(r,k)=1: A000040 (1,0), A065091 (2,1), A002476 (3,1), A003627 (3,2), A002144 (4,1), A002145 (4,3), A030430 (5,1), A045380 (5,2), A030431 (5,3), A030433 (5,4), A002476 (6,1), this sequence (6,5), A140444 (7,1), A045392 (7,2), A045437 (7,3), A045471 (7,4), A045458 (7,5), A045473 (7,6), A007519 (8,1), A007520 (8,3), A007521 (8,5), A007522 (8,7), A061237 (9,1), A061238 (9,2), A061239 (9,4), A061240 (9,5), A061241 (9,7), A061242 (9,8), A030430 (10,1), A030431 (10,3), A030432 (10,7), A030433 (10,9), A141849 (11,1), A090187 (11,2), A141850 (11,3), A141851 (11,4), A141852 (11,5), A141853 (11,6), A141854 (11,7), A141855 (11,8), A141856 (11,9), A141857 (11,10), A068228 (12,1), A040117 (12,5), A068229 (12,7), A068231 (12,11).
Cf. A034694 (smallest prime == 1 (mod n)).
Cf. A038700 (smallest prime == n-1 (mod n)).
Cf. A038026 (largest possible value of smallest prime == r (mod n)).
Cf. A001359 (lesser of twin primes), A005384 (Sophie Germain primes).
Cf. A048265, A324076.

Filtered(List([1..100],n->6*n-1),IsPrime); # Muniru A Asiru, May 19 2018

a007528 n = a007528_list !! (n-1)
a007528_list = [x | k <- [0..], let x = 6 * k + 5, a010051' x == 1]
-- Reinhard Zumkeller, Jul 13 2012

select(isprime,[seq(6*n-1,n=1..100)]); # Muniru A Asiru, May 19 2018

Select[6 Range[100]-1,PrimeQ]  (* Harvey P. Dale, Feb 14 2011 *)

forprime(p=2, 1e3, if(p%6==5, print1(p, ", "))) \\ Charles R Greathouse IV, Jul 15 2011

forprimestep(p=5,1000,6, print1(p", ")) \\ Charles R Greathouse IV, Mar 03 2025

A003627 \ {2}. - R. J. Mathar, Oct 28 2008
Conjecture: Product_{n >= 1} ((a(n) - 1) / (a(n) + 1)) * ((A002476(n) + 1) / (A002476(n) - 1)) = 3/4. - Dimitris Valianatos, Feb 11 2020
From Vaclav Kotesovec, May 02 2020: (Start)
Product_{k>=1} (1 - 1/a(k)^2) = 9*A175646/Pi^2 = 1/1.060548293.... =4/(3*A333240).
Product_{k>=1} (1 + 1/a(k)^2) = A334482.
Product_{k>=1} (1 - 1/a(k)^3) = A334480.
Product_{k>=1} (1 + 1/a(k)^3) = A334479. (End)
Legendre symbol (-3, a(n)) = -1 and (-3, A002476(n)) = +1, for n >= 1. For prime 3 one sets (-3, 3) = 0. - Wolfdieter Lang, Mar 03 2021

A068228 Primes congruent to 1 (mod 12).

Original entry on oeis.org

13, 37, 61, 73, 97, 109, 157, 181, 193, 229, 241, 277, 313, 337, 349, 373, 397, 409, 421, 433, 457, 541, 577, 601, 613, 661, 673, 709, 733, 757, 769, 829, 853, 877, 937, 997, 1009, 1021, 1033, 1069, 1093, 1117, 1129, 1153, 1201, 1213, 1237, 1249, 1297

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu), Feb 22 2002

This has several equivalent definitions (cf. the Tunnell link)
Also primes of the form x^2 + 9y^2 (discriminant -36). - T. D. Noe, May 07 2005 [corrected by Klaus Purath, Jan 18 2023]
Also primes of the form x^2 - 12y^2 (discriminant 48). Cf. A140633. - T. D. Noe, May 19 2008 [corrected by Klaus Purath, Jan 18 2023]
Also primes of the form x^2 + 4*x*y + y^2.
Also primes of the form x^2 + 2*x*y - 2*y^2 (cf. A084916).
Also primes of the form x^2 + 6*x*y - 3*y^2.
Also primes of the form 4*x^2 + 8*x*y + y^2.
Also primes of the form u^2 - 3v^2 (use the transformation {u,v} = {x+2y,y}). - Tito Piezas III, Dec 28 2008
Sequence lists generalized cuban primes (A007645) that are the sum of 2 nonzero squares. - Altug Alkan, Nov 25 2015
Yasutoshi Kohmoto observes that prevprime(a(n)) is more frequently congruent to 3 (mod 4) than to 1. This bias can be explained by the possible prime constellations and gaps: To have the same residue mod 4 as a prime in the list, the previous prime must be at a gap of 4 or 8 or 12 ..., but a gap of 4 is impossible because 12k + 1 - 4 is divisible by 3, and gaps >= 12 are very rare for small primes. To have the residue 3 (mod 4) the previous prime can be at a gap of 2 or 6 with no a priori divisibility property. However, this bias tends to disappear as the primes (and average prime gaps) grow bigger: for primes < 10^5, the ratio is about 35% vs. 65% as the above simple explanation suggests, but considering primes up to 10^8 yields a ratio of about 41% vs. 59%. It can be expected that the ratio asymptotically tends to 1:1. - M. F. Hasler, Sep 01 2017
Also primes of the form x^2 - 27*y^2. - Klaus Purath, Jan 18 2023

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.
David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
William C. Jagy and Irving Kaplansky, Positive definite binary quadratic forms that represent the same primes [Cached copy, with permission]
Michael Penn, an example right from my number theory class., YouTube video, 2021.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
J. B. Tunnell, Proofs of Conjectures Concerning Entry A033212, the Sequence of Primes Congruent to 1 or 19 (mod 30)
J. Voight, Quadratic forms that represent almost the same primes, Math. Comp., Vol. 76 (2007), pp. 1589-1617.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Cf. A068227, A068229, A040117, A068231, A068232, A068233, A068234, A068235, A139643, A141122, A140633, A264732.
Subsequence of A084916.
Subsequence of A007645.
Also primes in A084916, A020672.
Cf. A141123 (d=12), A141111, A141112 (d=65), A141187 (d=48) A038872 (d=5), A038873 (d=8), A038883 (d=13), A038889 (d=17).
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

[p: p in PrimesUpTo(1400) | p mod 12 in {1}]; // Vincenzo Librandi, Jul 14 2012
For other programs see the "Binary Quadratic Forms and OEIS" link.

select(isprime, [seq(i,i=1..10000, 12)]); # Robert Israel, Nov 27 2015

Select[Prime/@Range[250], Mod[ #, 12]==1&]
Select[Range[13, 10^4, 12], PrimeQ] (* Zak Seidov, Mar 21 2011 *)

for(i=1,250, if(prime(i)%12==1, print(prime(i))))

forstep(p=13,10^4,12,isprime(p)&print(p)); \\ Zak Seidov, Mar 21 2011

Edited by Dean Hickerson, Feb 27 2002

Entry revised by N. J. A. Sloane, Oct 18 2014 (Edited, merged with A141122, submitted by Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 05 2008).

A068231 Primes congruent to 11 mod 12.

Original entry on oeis.org

11, 23, 47, 59, 71, 83, 107, 131, 167, 179, 191, 227, 239, 251, 263, 311, 347, 359, 383, 419, 431, 443, 467, 479, 491, 503, 563, 587, 599, 647, 659, 683, 719, 743, 827, 839, 863, 887, 911, 947, 971, 983, 1019, 1031, 1091, 1103, 1151, 1163, 1187, 1223

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu), Feb 22 2002

Intersection of A002145 (primes of form 4n+3) and A003627 (primes of form 3n-1). So these are both Gaussian primes with no imaginary part and Eisenstein primes with no imaginary part. - Alonso del Arte, Mar 29 2007
Is this the same sequence as A141187 (apart from the initial 3)?
If p is prime of the form 2*a(n)^k + 1, then p divides a cyclotomic number Phi(a(n)^k, 2). - Arkadiusz Wesolowski, Jun 14 2013
Also a(n) = primes p dividing A014138((p-3)/2), where A014138(n) = Partial sums of (Catalan numbers starting 1,2,5,...), cf. A000108. - Alexander Adamchuk, Dec 27 2013

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

Cf. A068227, A068228, A068229, A040117, A068232, A068233, A068234, A068235, A000040, A014138, A000108.

%4n-1 and 6n-1 primes
n = 1:10000;
n2 = 4*n-1;
n3 = 3*n-1;
p = primes(max(n2));
Res = intersect(n2,n3);
Res2 = intersect(Res,p);
% Jesse H. Crotts, Sep 25 2016

[p: p in PrimesUpTo(1500) | p mod 12 eq 11 ]; // Vincenzo Librandi, Aug 14 2012

Select[Prime/@Range[250], Mod[ #, 12]==11&]
Select[Range[11,1500,12],PrimeQ] (* Harvey P. Dale, Sep 15 2023 *)

for(i=1,250, if(prime(i)%12==11, print(prime(i))))

Edited by Dean Hickerson, Feb 27 2002

A068229 Primes congruent to 7 (mod 12).

Original entry on oeis.org

7, 19, 31, 43, 67, 79, 103, 127, 139, 151, 163, 199, 211, 223, 271, 283, 307, 331, 367, 379, 439, 463, 487, 499, 523, 547, 571, 607, 619, 631, 643, 691, 727, 739, 751, 787, 811, 823, 859, 883, 907, 919, 967, 991, 1039, 1051, 1063, 1087, 1123, 1171, 1231

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu), Feb 22 2002

Primes of the form 3x^2 + 4y^2. - T. D. Noe, May 08 2005

It appears that all terms starting from term 103 are primes which are the sum of 5 positive (n > 0) different squares in more than one way (A193143) - Vladimir Joseph Stephan Orlovsky, Jul 16 2011.

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

Cf. A068227, A068228, A040117, A068231, A068232, A068233, A068234, A068235.

[ p: p in PrimesUpTo(1400) | p mod 12 in {7} ]; // Vincenzo Librandi, Jul 14 2012

Select[Prime/@Range[250], Mod[#, 12] == 7 &]

for(i=1,250, if(prime(i)%12==7, print(prime(i))))

is_A068229(n)=n%12==7 && isprime(n) \\ then, e.g.,
select(is_A068229, primes(250))  \\ - M. F. Hasler, Jan 25 2013

a(n) ~ 4n log n. - Charles R Greathouse IV, Dec 07 2022

Edited by Dean Hickerson, Feb 27 2002

A145471 Primes p such that (5+p)/2 is prime.

Original entry on oeis.org

5, 17, 29, 41, 53, 89, 101, 113, 137, 173, 197, 257, 269, 293, 353, 389, 449, 461, 509, 521, 557, 617, 701, 761, 773, 797, 857, 881, 929, 953, 977, 1013, 1109, 1181, 1193, 1229, 1277, 1289, 1301, 1361, 1433, 1481, 1613, 1637, 1709, 1721, 1877, 1889, 1901

Offset: 1

Artur Jasinski, Oct 11 2008

nonn

All these primes are congruent to 1 mod 4 and to 5 mod 12.

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

Cf. A063909, A092109.

Subsequence of A040117. - Zak Seidov, Feb 21 2016

[p: p in PrimesInInterval(3,2000) | IsPrime((5+p) div 2)]; // Vincenzo Librandi, Feb 25 2016

select(t -> isprime(t) and isprime((t+5)/2), [seq(i, i=5..1000, 12)]); # Robert Israel, Feb 24 2016

aa = {}; k = 5; Do[If[PrimeQ[(k + Prime[n])/2], AppendTo[aa, Prime[n]]], {n, 1, 500}];aa
Select[Prime[Range[500]],PrimeQ[(5+#)/2]&]  (* Harvey P. Dale, Apr 23 2011 *)

forprime(p=2,1e4,if(p%12!=5,next);if(isprime(p\2+3),print1(p", "))) \\ Charles R Greathouse IV, Jul 16 2011

a(n) = 2*A063909(n)-5. - Robert Israel, Feb 24 2016

A154577 Primes of the form 2n^2+14n+5.

Original entry on oeis.org

5, 41, 293, 401, 461, 593, 821, 1181, 1493, 1721, 3593, 4493, 5081, 7793, 12941, 16001, 16361, 17093, 21821, 28541, 29021, 31481, 33521, 36161, 39461, 45281, 48341, 54101, 56093, 65141, 66593, 74093, 75641, 76421, 83621, 92861, 98993, 101681

Offset: 1

Vincenzo Librandi, Jan 12 2009

Primes in A154576. [Omar E. Pol, Aug 05 2009]
Subsequence of A007528, A040117. [Bruno Berselli, Jun 18 2014]
2*a(n) + 39 is a square. - Vincenzo Librandi, Apr 10 2015

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

Cf. A007528, A040117, A154576.

[a: n in [0..300] | IsPrime(a) where a is 2*n^2+14*n+5]; // Vincenzo Librandi, Jul 23 2012

Select[Table[2n^2+14n+5,{n,0,15001}],PrimeQ] (* Vincenzo Librandi, Jul 23 2012 *)

for (n=0, 300, if (isprime (k=2*n^2+14*n+5), print1 (k, ",  "))); \\ Vincenzo Librandi, Jul 23 2012

a(30)=66593 inserted, definition corrected and edited by Omar E. Pol, Aug 05 2009, Aug 06 2009

Added the first term 5 by Vincenzo Librandi, Jul 23 2012

A141849 Primes congruent to 1 mod 11.

Original entry on oeis.org

23, 67, 89, 199, 331, 353, 397, 419, 463, 617, 661, 683, 727, 859, 881, 947, 991, 1013, 1123, 1277, 1321, 1409, 1453, 1607, 1783, 1871, 2003, 2069, 2113, 2179, 2267, 2311, 2333, 2377, 2399, 2531, 2663, 2707, 2729, 2861, 2927, 2971, 3037, 3169, 3191, 3257

Offset: 1

N. J. A. Sloane, Jul 11 2008

Conjecture: Also primes p such that ((x+1)^11-1)/x has 10 distinct irreducible factors of degree 1 over GF(p). - Federico Provvedi, Apr 17 2018

Primes congruent to 1 mod 22. - Chai Wah Wu, Apr 28 2025

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

Cf. A000040, A090187, A102656.
Prime sequences A# (k,r) of the form k*n+r with 0 <= r <= k-1 (i.e., primes == r (mod k), or primes p with p mod k = r) and gcd(r,k)=1: A000040 (1,0), A065091 (2,1), A002476 (3,1), A003627 (3,2), A002144 (4,1), A002145 (4,3), A030430 (5,1), A045380 (5,2), A030431 (5,3), A030433 (5,4), A002476 (6,1), A007528 (6,5), A140444 (7,1), A045392 (7,2), A045437 (7,3), A045471 (7,4), A045458 (7,5), A045473 (7,6), A007519 (8,1), A007520 (8,3), A007521 (8,5), A007522 (8,7), A061237 (9,1), A061238 (9,2), A061239 (9,4), A061240 (9,5), A061241 (9,7), A061242 (9,8), A030430 (10,1), A030431 (10,3), A030432 (10,7), A030433 (10,9), this sequence (11,1), A090187 (11,2), A141850 (11,3), A141851 (11,4), A141852 (11,5), A141853 (11,6), A141854 (11,7), A141855 (11,8), A141856 (11,9), A141857 (11,10), A068228 (12,1), A040117 (12,5), A068229 (12,7), A068231 (12,11).
Cf. A034694 (smallest prime == 1 (mod n)).
Cf. A038700 (smallest prime == n-1 (mod n)).
Cf. A038026 (largest possible value of smallest prime == r (mod n)).

Filtered([1..4000],n->n mod 11=1 and IsPrime(n)); # Muniru A Asiru, Apr 19 2018

[ p: p in PrimesUpTo(5000) | p mod 11 eq 1 ]; // Vincenzo Librandi, Apr 19 2011

a:=select(n->isprime(n) and modp(n,11)=1,[$1..4000]); # Muniru A Asiru, Apr 19 2018

Select[Range[1,10000,11],PrimeQ] (* Vladimir Joseph Stephan Orlovsky, May 18 2011 *)

is(n)=isprime(n) && n%11==1 \\ Charles R Greathouse IV, Jul 01 2016

forstep(n=2, 1e3, 2, if(isprime(p=11*n+1), print1(p, ", "))); \\ Altug Alkan, Apr 19 2018

a(n) ~ 10n log n. - Charles R Greathouse IV, Jul 02 2016

A068227 The "genity" sequence of the primes, i.e., a(n) = g(p) = ((p mod 4) + (p mod 6))/2, where p is the n-th prime.

Original entry on oeis.org

2, 3, 3, 2, 4, 1, 3, 2, 4, 3, 2, 1, 3, 2, 4, 3, 4, 1, 2, 4, 1, 2, 4, 3, 1, 3, 2, 4, 1, 3, 2, 4, 3, 2, 3, 2, 1, 2, 4, 3, 4, 1, 4, 1, 3, 2, 2, 2, 4, 1, 3, 4, 1, 4, 3, 4, 3, 2, 1, 3, 2, 3, 2, 4, 1, 3, 2, 1, 4, 1, 3, 4, 2, 1, 2, 4, 3, 1, 3, 1, 4, 1, 4, 1, 2, 4, 3, 1, 3, 2, 4, 4, 2, 4, 2, 4, 3, 3, 2, 1, 2, 3, 4

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu), Feb 22 2002

The name "genity" was derived from "genes" and "parity", since the fourfold values of g(p) in a sequence corresponding to prime arguments resemble the genetic sequences of the nucleotides in the DNA. Parity is also related, since it originally means a (mod 2) feature, while here we categorize the primes (mod 4) and (mod 6), simultaneously.
The arithmetic function g(p) = ((p mod 4) + (p mod 6))/2 provides integer values for prime arguments, such that 1 <= g(p) <= 4 and is determined by the congruence class of p (mod 12). Specifically, g(p)=1 if p==1 (mod 12), g(p)=2 if p=2 or p==7 (mod 12), g(p)=3 if p=3 or p==5 (mod 12) and g(p)=4 if p==11 (mod 12).
Dickson's conjecture implies that every finite sequence of numbers from 1 to 4 occurs infinitely often in this sequence.

Robert Price, Table of n, a(n) for n = 1..10000
The Prime Glossary, Dickson's conjecture

Cf. A068228, A068229, A040117, A068231, A068232, A068233, A068234, A068235.

Table[(Mod[Prime[n], 4] + Mod[Prime[n], 6])/2, {n, 1, 100}]

for(i=1,120,print((prime(i)%4+prime(i)%6)/2))

Edited by Dean Hickerson and Robert G. Wilson v, Mar 06 2002

Showing 1-10 of 28 results. Next

cp's OEIS Frontend

A068230 Duplicate of A040117.

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A106856 Primes of the form x^2 + xy + 2y^2, with x and y nonnegative.

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A007528 Primes of the form 6k-1.

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A068228 Primes congruent to 1 (mod 12).

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A068231 Primes congruent to 11 mod 12.

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A068229 Primes congruent to 7 (mod 12).

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A145471 Primes p such that (5+p)/2 is prime.

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