A068231 -id:A068231 - cp's OEIS frontend

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.

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

A068233 a(n) is the smallest prime p such that p and the next n-1 primes are all == 7 (mod 12).

Original entry on oeis.org

7, 199, 199, 32443, 180799, 180799, 4338787, 84885631, 472798219, 1786054267, 6024282871, 64791932287, 592175010019, 6265824724519, 7816088451907, 24660781037467

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, A068232, A068234, A068235.

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) 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

A116610 Values of n such that prime(2*n) mod 12 = 11.

Original entry on oeis.org

10, 14, 16, 26, 27, 28, 32, 36, 38, 43, 46, 47, 48, 59, 60, 62, 64, 66, 72, 73, 75, 77, 78, 82, 83, 91, 95, 96, 100, 104, 107, 114, 115, 118, 120, 123, 124, 125, 128, 131, 140, 143, 146, 147, 152, 159, 167, 168, 173, 179, 180, 182, 185, 186, 188, 193, 195, 205, 210

Offset: 1

Roger L. Bagula, Mar 29 2006

nonn

			26 is in the sequence because the 52nd prime is 239 and 239 mod 12=11.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A116602, A116612-A116617, A160591-A160594. - M. F. Hasler, May 22 2009

a:=proc(n) if ithprime(2*n) mod 12 = 11 then n else fi end: seq(a(n),n=1..250);

Select[Range[250],Mod[Prime[2#],12]==11&]  (* Harvey P. Dale, Jan 30 2011 *)

for(n=1,999, prime(2*n)%12==11 & print1(n",")) \\ M. F. Hasler, May 22 2009

A116610 = 1/2 { even terms in A160593 = A000720(A068231) } . - M. F. Hasler, May 22 2009

Edited by N. J. A. Sloane, Apr 05 2006

A141187 Primes of the form -x^2+6xy+3y^2 (as well as of the form 8x^2+12xy+3*y^2).

Original entry on oeis.org

3, 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

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (sergarmor(AT)yahoo.es), Jun 12 2008

nonn

Discriminant = 48. Class = 2. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac and gcd(a,b,c)=1.
Values of the quadratic form are {0,3,8,11} mod 12, so all values with the exception of 3 are also in A068231. - R. J. Mathar, Jul 30 2008
Is this the same sequence (apart from the initial 3) as A068231? [Yes, since the orders of imaginary quadratic fields with discriminant 48 has 1 class per genus (can be verified by the quadclassunit() function in PARI), so the primes represented by a binary quadratic form of this discriminant are determined by a congruence condition. - Jianing Song, Jun 22 2025]

			a(3)=23 because we can write 23= -1^2+6*1*2+3*2^2 (or 23=8*1^2+12*1*1+3*1^2).

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A038872 (d=5), A038873 (d=8), A068228 (d=12, 48, or -36), A038883 (d=13), A038889 (d=17), A141111 and A141112 (d=65).
Essentially the same as A068231 and A141123.
Cf. A243169.
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

Reap[For[p = 2, p < 2000, p = NextPrime[p], If[FindInstance[p == -x^2 + 6*x*y + 3*y^2, {x, y}, Integers, 1] =!= {}, Print[p]; Sow[p]]]][[2, 1]] (* Jean-François Alcover, Oct 25 2016 *)

More terms from Colin Barker, Apr 05 2015

A243888 Primes of the form 2n^2+26n+11.

Original entry on oeis.org

71, 107, 191, 239, 347, 1031, 1439, 1667, 1787, 2039, 2447, 2591, 3371, 3539, 5231, 5651, 5867, 6311, 7247, 9311, 9587, 10151, 11027, 11939, 12251, 14207, 14891, 19727, 20939, 21767, 23039, 27539, 30431, 34511, 36107, 39971, 41687, 46439, 47051, 56039, 56711

Offset: 1

Vincenzo Librandi, Jun 16 2014

Subsequence of A068231.
Conjecture: except 107, 2^a(n)-1 is not prime; in other words, these primes are included in A054723.
2*a(n) + 147 is a square. - Vincenzo Librandi, Apr 10 2015

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

Cf. A068231.

Cf. Primes of the form 2*n^2+2*(2*k+3)*n+(2*k+1): A176549 (k=0), A221902 (k=1), A154577 (k=2), A154592 (k=3), A154601 (k=4), this sequence (k=5), A243889 (k=6), A217494 (k=7), A243890 (k=8), A221903 (k=9), A217495 (k=10), A217496 (k=11), A217497 (k=12), A217498 (k=13), A243891 (k=14), A243957 (k=15), A217499 (k=16), A217500 (k=17), A217501 (k=18), A217620 (k=19), A243958 (k=20), A217621 (k=21).

[a: n in [1..200] | IsPrime(a) where a is 2*n^2+26*n+11];

Select[Table[2 n^2 + 26 n + 11, {n, 800}], PrimeQ]

A040101 Primes p such that x^4 = 3 has a solution mod p.

Original entry on oeis.org

2, 3, 11, 13, 23, 47, 59, 71, 83, 107, 109, 131, 167, 179, 181, 191, 193, 227, 229, 239, 251, 263, 277, 311, 313, 347, 359, 383, 419, 421, 431, 433, 443, 467, 479, 491, 503, 541, 563, 577, 587, 599, 601, 647, 659

Offset: 1

N. J. A. Sloane

Union of 2, 3, A068231 (primes congruent to 11 modulo 12), primes p == 1 (mod 4) such that 3^((p-1)/4) == 1 (mod p). - Jianing Song, Jun 22 2025

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

A subsequence of A038874.

A068231 < A385220 < A045317 < this sequence < A097933 (ignoring terms 2, 3), where Ax < Ay means that Ax is a subsequence of Ay.

[p: p in PrimesUpTo(800) | exists(t){x : x in ResidueClassRing(p) | x^4 eq 3}]; // Vincenzo Librandi, Sep 11 2012

ok [p_]:=Reduce[Mod[x^4- 3, p] == 0, x, Integers] =!= False;  Select[Prime[Range[200]], ok] (* Vincenzo Librandi, Sep 11 2012 *)

isA040101(p) = isprime(p) && (p==2 || p==3 || p%12==11 || (p%4==1 && Mod(3, p)^((p-1)/4) == 1)) \\ Jianing Song, Jun 22 2025

A045317 Primes p such that x^8 = 3 has a solution mod p.

Original entry on oeis.org

2, 3, 11, 13, 23, 47, 59, 71, 83, 107, 109, 131, 167, 179, 181, 191, 227, 229, 239, 251, 263, 277, 311, 313, 347, 359, 383, 419, 421, 431, 433, 443, 467, 479, 491, 503, 541, 563, 587, 599, 601, 647, 659, 683, 709

Offset: 1

N. J. A. Sloane

Complement of A045318 relative to A000040. - Vincenzo Librandi, Sep 13 2012

Union of 2, 5, A068231 (primes congruent to 11 modulo 12), prime p == 5 (mod 8) such that 3^((p-1)/4) == 1 (mod p), and primes p == 1 (mod 8) such that 3^((p-1)/8) == 1 (mod p). - Jianing Song, Jun 22 2025

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

Cf. A000040, A045318.

A068231 < A385220 < this sequence < A040101 < A097933 (ignoring terms 2, 3), where Ax < Ay means that Ax is a subsequence of Ay.

[p: p in PrimesUpTo(800) | exists(t){x : x in ResidueClassRing(p) | x^8 eq 3}]; // Vincenzo Librandi, Sep 13 2012

ok[p_]:= Reduce[Mod[x^8- 3, p] == 0, x, Integers]=!=False; Select[Prime[Range[200]], ok] (* Vincenzo Librandi, Sep 13 2012 *)

isok(p) = isprime(p) && ispower(Mod(3, p), 8); \\ Michel Marcus, Oct 17 2018

isA045317(p) = isprime(p) && (p==2 || p==3 || p%12==11 || (p%8==5 && Mod(3, p)^((p-1)/4) == 1) || (p%8==1 && Mod(3, p)^((p-1)/8) == 1)) \\ Jianing Song, Jun 22 2025

cp's OEIS Frontend

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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A068233 a(n) is the smallest prime p such that p and the next n-1 primes are all == 7 (mod 12).

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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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PARI

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

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Programs

Mathematica

PARI

Extensions

A116610 Values of n such that prime(2*n) mod 12 = 11.

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Maple

Mathematica

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Extensions

A141187 Primes of the form -x^2+6*x*y+3*y^2 (as well as of the form 8*x^2+12*x*y+3*y^2).

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Links

Crossrefs

Programs

Mathematica

A141187 Primes of the form -x^2+6xy+3y^2 (as well as of the form 8x^2+12xy+3*y^2).

A243888 Primes of the form 2n^2+26n+11.