A068233 -id:A068233 - cp's OEIS frontend

A068228 Primes congruent to 1 (mod 12).

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

A040117 Primes congruent to 5 (mod 12). Also primes p such that x^4 = 9 has no solution mod p.

Original entry on oeis.org

5, 17, 29, 41, 53, 89, 101, 113, 137, 149, 173, 197, 233, 257, 269, 281, 293, 317, 353, 389, 401, 449, 461, 509, 521, 557, 569, 593, 617, 641, 653, 677, 701, 761, 773, 797, 809, 821, 857, 881, 929, 941, 953, 977, 1013, 1049, 1061, 1097, 1109, 1181, 1193

Offset: 1

N. J. A. Sloane

Odd primes of the form 2x^2-2xy+5y^2 with x and y nonnegative. - T. D. Noe, May 08 2005, corrected by M. F. Hasler, Jul 03 2025
Complement of A040116 relative to A000040. - Vincenzo Librandi, Sep 17 2012
Odd primes of the form a^2 + b^2 such that a^2 == b^2 (mod 3). - Thomas Ordowski and Charles R Greathouse IV, May 20 2015
Yasutoshi Kohmoto observes that nextprime(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 next prime must be at a gap of 4 or 8 or 12..., but a gap of 4 is impossible because 12k + 5 + 4 is divisible by 3, and gaps >= 12 are very rare for small primes. To have the residue 3 (mod 4) the next 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 40% vs 60%. It can be expected that the ratio asymptotically tends to 1:1. - M. F. Hasler, Sep 01 2017

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

Cf. A068227, A068228, A068229, A068231, A068232, A068233, A068234, A068235.

Equal to A243183 (primes of the form 2x^2+2xy+5y^2) except for the additional A243183(1) = 2 (and indexing of subsequent terms).

[p: p in PrimesUpTo(1200) | not exists{x : x in ResidueClassRing(p) | x^4 eq 9} ]; // Vincenzo Librandi, Sep 17 2012

Select[Prime/@Range[250], Mod[ #, 12]==5&]
ok[p_]:= Reduce[Mod[x^4 - 9, p] == 0, x, Integers] == False;Select[Prime[Range[200]], ok] (* Vincenzo Librandi, Sep 17 2012 *)

for(i=1,250, if(prime(i)%12==5, print(prime(i))))

a(n) ~ 4n log n. - Charles R Greathouse IV, May 20 2015

More terms from Dean Hickerson, Feb 27 2002

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

A068232 a(n) is the smallest prime p such that p and the next n-1 primes are all == 1 (mod 12).

Original entry on oeis.org

13, 661, 8317, 12829, 586153, 1081417, 7790917, 7790917, 370861009, 370861009, 370861009, 5637496849, 289391626057, 469257742237, 628337233501, 84424712545429, 155494152002017, 341821313785729

Offset: 1

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

Dickson's conjecture implies that a(n) exists for all n.

The Prime Glossary, Dickson's conjecture

Cf. A068227, A068228, A068229, A040117, A068231, A068233, A068234, A068235.

For[i=n=1, True, Null, For[j=0, jHarvey P. Dale, Dec 24 2020 *)

PARI
```
{i=n=1; while(1,j=0; while(j
				
```

Edited by Dean Hickerson, Mar 06 2002
a(12)-a(15) from Giovanni Resta, Feb 18 2006
a(16)-a(18) from Giovanni Resta, Aug 04 2013

A068234 a(n) is the smallest prime p such that p and the next n-1 primes are all == 5 (mod 12).

Original entry on oeis.org

5, 509, 4397, 42509, 647417, 647417, 1248869, 13175609, 234946997, 1039154933, 7114719473, 32021552837, 32021552837, 1237381737257, 2904797643617, 2904797643617, 2904797643617

Offset: 1

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

Dickson's conjecture implies that a(n) exists for all n.

a(18) > 4*10^14. - Giovanni Resta, Aug 04 2013

The Prime Glossary, Dickson's conjecture

Cf. A068227, A068228, A068229, A040117, A068231, A068232, A068233, A068235.

For[i=n=1, True, Null, For[j=0, jHarvey P. Dale, Feb 02 2022 *)

PARI
```
{i=n=1; while(1,j=0; while(j
				
```

Edited by Dean Hickerson, Mar 06 2002

More terms from Giovanni Resta, Feb 18 2006

A068235 a(n) is the smallest prime p such that p and the next n-1 primes are all == 11 (mod 12).

Original entry on oeis.org

11, 467, 1499, 16763, 260339, 2003387, 7722419, 20221283, 927161471, 4284484931, 7355362139, 84805717127, 478527373859, 2046207697631, 7302359785151, 21104656617827, 21104656617827

Offset: 1

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

Dickson's conjecture implies that a(n) exists for all n.

a(18) > 4*10^14. - Giovanni Resta, Aug 04 2013

The Prime Glossary, Dickson's conjecture

Cf. A068227, A068228, A068229, A040117, A068231, A068232, A068233, A068234.

Mathematica
```
For[i=n=1, True, Null, For[j=0, j
				
```
PARI
```
{i=n=1; while(1,j=0; while(j
				
```

Edited by Dean Hickerson, Mar 06 2002
More terms from Giovanni Resta, Feb 18 2006
a(16)-a(17) from Giovanni Resta, Aug 04 2013

cp's OEIS Frontend

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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A040117 Primes congruent to 5 (mod 12). Also primes p such that x^4 = 9 has no solution mod p.

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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.

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A068232 a(n) is the smallest prime p such that p and the next n-1 primes are all == 1 (mod 12).

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Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A068234 a(n) is the smallest prime p such that p and the next n-1 primes are all == 5 (mod 12).

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