A068228 -id:A068228 - cp's OEIS frontend

A110801 Numbers n such that 12n + 1 is prime.

1, 3, 5, 6, 8, 9, 13, 15, 16, 19, 20, 23, 26, 28, 29, 31, 33, 34, 35, 36, 38, 45, 48, 50, 51, 55, 56, 59, 61, 63, 64, 69, 71, 73, 78, 83, 84, 85, 86, 89, 91, 93, 94, 96, 100, 101, 103, 104, 108, 110, 115, 119, 121, 124, 129, 133, 134, 135, 138, 139, 141, 145, 146, 148

Offset: 1

Parthasarathy Nambi, Oct 20 2005

nonn

Corresponds to even numbers in A024898. - Michael B. Porter, Oct 27 2009

			If n=96 then 12*n + 1 = 1153 (prime).

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

Cf. A167055, A167056, A167057, A024898; primes are in A068228. - Michael B. Porter, Oct 27 2009

[ n: n in [1..150] | IsPrime(12*n+1) ]; // Klaus Brockhaus, Jan 02 2009

Select[Range[150],PrimeQ[12#+1]&] (* Harvey P. Dale, Jul 17 2018 *)

isA110801(n) = isprime(12*n+1) \\ Michael B. Porter, Oct 27 2009

More terms from Klaus Brockhaus, Jan 02 2009

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

A141373 Primes of the form 3x^2+16y^2. Also primes of the form 4x^2+4xy-5y^2 (as well as primes the form 4x^2+12xy+3y^2).

Original entry on oeis.org

3, 19, 43, 67, 139, 163, 211, 283, 307, 331, 379, 499, 523, 547, 571, 619, 643, 691, 739, 787, 811, 859, 883, 907, 1051, 1123, 1171, 1291, 1459, 1483, 1531, 1579, 1627, 1699, 1723, 1747, 1867, 1987, 2011, 2083, 2131, 2179, 2203, 2251, 2347, 2371, 2467, 2539

Offset: 1

T. D. Noe, May 13 2005; Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 28 2008

nonn

The discriminant is -192 (or 96, or ...), depending on which quadratic form is used for the definition. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac and gcd(a,b,c)=1. See A107132 for more information.

Except for 3, also primes of the forms 4x^2 + 4xy + 19y^2 and 16x^2 + 8xy + 19y^2. See A140633. - T. D. Noe, May 19 2008

			19 is a member because we can write 19=4*2^2+4*2*1-5*1^2 (or 19=4*1^2+12*1*1+3*1^2).

Z. I. Borevich and I. R. Shafarevich, Number Theory.

Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
William C. Jagy and Irving Kaplansky, Positive definite binary quadratic forms that represent the same primes [Cached copy] See Table II.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS: Index to related sequences, programs, references. OEIS wiki, June 2014.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Cf. A107132, A139827, A107003, A141375, A141376 (d=96).
See also A038872 (d=5),
A038873 (d=8),
A068228, A141123 (d=12),
A038883 (d=13),
A038889 (d=17),
A141158 (d=20),
A141159, A141160 (d=21),
A141170, A141171 (d=24),
A141172, A141173 (d=28),
A141174, A141175 (d=32),
A141176, A141177 (d=33),
A141178 (d=37),
A141179, A141180 (d=40),
A141181 (d=41),
A141182, A141183 (d=44),
A033212, A141785 (d=45),
A068228, A141187 (d=48),
A141188 (d=52),
A141189 (d=53),
A141190, A141191 (d=56),
A141192, A141193 (d=57),
A107152, A141302, A141303, A141304 (d=60),
A141215 (d=61),
A141111, A141112 (d=65),
A141336, A141337 (d=92),
A141338, A141339 (d=93),
A141161, A141163 (d=148),
A141165, A141166 (d=229),
A141167, A141167, A141167 (d=257).

[3] cat [ p: p in PrimesUpTo(3000) | p mod 24 in {19 } ]; // Vincenzo Librandi, Jul 24 2012

QuadPrimes2[3, 0, 16, 10000] (* see A106856 *)

list(lim)=my(v=List(),w,t); for(x=1, sqrtint(lim\3), w=3*x^2; for(y=0, sqrtint((lim-w)\16), if(isprime(t=w+16*y^2), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Feb 09 2017

Except for 3, the primes are congruent to 19 (mod 24). - T. D. Noe, May 02 2008

More terms from Colin Barker, Apr 05 2015

Edited by N. J. A. Sloane, Jul 14 2019, combining two identical entries both with multiple cross-references.

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

A141161 Primes of the form 4x^2+6xy-7y^2.

Original entry on oeis.org

3, 7, 11, 41, 47, 53, 71, 73, 83, 101, 127, 149, 157, 173, 181, 197, 211, 223, 229, 263, 271, 307, 337, 359, 373, 379, 397, 419, 433, 443, 509, 521, 571, 593, 599, 613, 617, 619, 641, 659, 673, 677, 719, 733, 739, 743, 751, 761, 773, 787, 811, 821, 887, 937

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 13 2008

nonn

Discriminant = 148. Class = 3. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac and gcd(a,b,c)=1.

Also primes represented by the improperly equivalent form 7*x^2 + 6*x*y - 4*y^2

			a(8)=73 because we can write 73= 4*4^2+6*4*3-7*3^2.

Z. I. Borevich and I. R. Shafarevich, Number Theory.

Juan Arias-de-Reyna, Table of n, a(n) for n = 1..10000
Peter Luschny, Binary Quadratic Forms
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS: Index to related sequences, programs, references. OEIS wiki, June 2014.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Cf. A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65). A141163 (d=148).

For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

q := 4*x^2 + 6*x*y - 7*y^2; pmax = 1000; xmax = 100; ymin = -xmax; ymax = xmax; k = 1; prms0 = {}; prms = {2}; While[prms != prms0, xx = yy = {}; prms0 = prms; prms = Reap[Do[p = q; If[2 <= p <= pmax && PrimeQ[p], AppendTo[xx, x]; AppendTo[yy, y]; Sow[p]], {x, 1, k*xmax}, {y, k*ymin, k*ymax}]][[2, 1]] // Union; xmax = Max[xx]; ymin = Min[yy]; ymax = Max[yy]; k++; Print["k = ", k, " xmax = ", xmax, " ymin = ", ymin, " ymax = ", ymax ]]; A141161 = prms (* Jean-François Alcover, Oct 26 2016 *)

# The function binaryQF is defined in the link 'Binary Quadratic Forms'.
Q = binaryQF([4, 6, -7])
Q.represented_positives(937, 'prime')  # Peter Luschny, Oct 26 2016

A141165 Primes of the form 9x^2+7xy-5y^2.

Original entry on oeis.org

3, 5, 11, 17, 19, 43, 61, 71, 83, 97, 103, 149, 151, 167, 181, 233, 271, 277, 293, 307, 311, 337, 367, 373, 397, 401, 409, 421, 431, 433, 457, 463, 467, 491, 557, 569, 587, 631, 641, 661, 673, 683, 701, 733, 743, 751, 757, 769, 787, 821, 859, 863, 883, 911

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 = 229. Class = 3. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac. They can represent primes only if gcd(a,b,c)=1. [Edited by M. F. Hasler, Jan 27 2016]
Also primes represented by the improperly equivalent form 5*x^2+7*x*y-9*y^2. - Juan Arias-de-Reyna, Mar 17 2011
36*a(n) has the form z^2 - 229*y^2, where z = 18*x+7*y. [Bruno Berselli, Jun 25 2014]
Appears to be the complement of A141166 in A268155, primes that are squares mod 229. - M. F. Hasler, Jan 27 2016

			a(10)=97 because we can write 97= 9*3^2+7*3*1-5*1^2

Z. I. Borevich and I. R. Shafarevich, Number Theory
D. B. Zagier, Zetafunktionen und quadratische Körper

Juan Arias-de-Reyna, Table of n, a(n) for n = 1..10000
Peter Luschny, Binary Quadratic Forms
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, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65). A141166 (d=229).

For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

q := 9*x^2 + 7*x*y - 5*y^2; pmax = 1000; xmax = xmax0 = 50; ymin = ymin0 = -50; ymax = ymax0 = 50; k = 1.3 (expansion coeff. for maxima *); prms0 = {}; prms = {2}; While[prms != prms0, xx = yy = {}; prms0 = prms; prms = Reap[Do[p = q; If[2 <= p <= pmax && PrimeQ[p], AppendTo[xx, x]; AppendTo[yy, y]; Sow[p]], {x, 1, If[xmax == xmax0, xmax, Floor[k*xmax]]}, {y, If[ymin == ymin0, ymin, Floor[k*ymin]], If[ymax == ymax0, ymax, Floor[k*ymax]]}]][[2, 1]] // Union; xmax = Max[xx]; ymin = Min[yy]; ymax = Max[yy]; Print[Length[prms], " terms", "  xmax = ", xmax, "  ymin = ", ymin, "  ymax = ", ymax ]]; A141165 = prms (* Jean-François Alcover, Oct 26 2016 *)

is_A141165(p)=qfbsolve(Qfb(9,7,-5),p) \\ Returns nonzero (actually, a solution [x,y]) iff p is a member of the sequence. For efficiency it is assumed that p is prime. - M. F. Hasler, Jan 27 2016

# uses[binaryQF]
# The function binaryQF is defined in the link 'Binary Quadratic Forms'.
Q = binaryQF([9, 7, -5])
print(Q.represented_positives(911, 'prime')) # Peter Luschny, Oct 26 2016

cp's OEIS Frontend

A110801 Numbers n such that 12n + 1 is prime.

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A141849 Primes congruent to 1 mod 11.

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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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A141373 Primes of the form 3*x^2+16*y^2. Also primes of the form 4*x^2+4*x*y-5*y^2 (as well as primes the form 4*x^2+12*x*y+3*y^2).

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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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Mathematica

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Extensions

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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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).

Views

Author

A141373 Primes of the form 3x^2+16y^2. Also primes of the form 4x^2+4xy-5y^2 (as well as primes the form 4x^2+12xy+3y^2).

A141161 Primes of the form 4x^2+6xy-7y^2.

A141165 Primes of the form 9x^2+7xy-5y^2.