A068231 -id:A068231 - cp's OEIS frontend

A270596 Primes p congruent to 11 mod 12 (A068231), such that there exists a nonzero element c of GF(p), such that the element c, c-1 and -1 generate a proper subgroup of the multiplicative group.

131, 191, 239, 251, 311, 419, 431, 491, 599, 647, 659, 683, 743, 827, 911, 971, 1031, 1091, 1103, 1151, 1163, 1223, 1259, 1451, 1499, 1511, 1559, 1571, 1583, 1607, 1667, 1787, 1811, 1847, 1871, 1931, 2003, 2087, 2111, 2243, 2267, 2339, 2351, 2399, 2411, 2423, 2531, 2591, 2663, 2687, 2699, 2711, 2843, 2927, 2939, 3011

Offset: 1

Michel Marcus, Mar 20 2016

nonn

P. Cameron shows that "primes congruent to 1 (mod 3) and greater than 7" (see A002476) and "primes congruent to 1 (mod 4) and greater than 5" (see A002144) also have this property.

Joerg Arndt, Table of n, a(n) for n = 1..1983
Peter Cameron's Blog, Permutation groups and regular semigroups, 2, Posted 22/08/2015.
Michel Marcus, GAP program with issues

Cf. A002476, A002144, A068231.

{ forprime(p=11, 10^6,
    if ( p%12 != 11, next() );
    for (c=2, p-2,
        my( v = vector(p-1) );
        my( g0 = Mod(c, p),  rc0 = znorder(g0) );
        if ( rc0 == p - 1,  next() );
        if ( znorder( -g0 ) == p - 1,  next() );
        my( g1 = Mod(c-1, p),  rc1 = znorder(g1) );
        if ( rc1 == p - 1,  next() );
        if ( znorder( -g1 ) == p - 1,  next() );
        if ( znorder( g0*g1 ) == p - 1,  next() );
        if ( znorder( -g0*g1 ) == p - 1,  next() );
        for (x0 = 0, rc0,
            my ( p0 = g0^x0,  z = p0 );
            for (x1 = 0, rc1,
                v[lift(z)] = 1;
                v[p - lift(z)] = 1;
                z * = g1;
            );
        );
        my( s = sum(k=1,#v,v[k]) );
        if ( s < p - 1,  print1(p,", "); break() );
    );
); } \\ Joerg Arndt, Mar 20 2016

Terms > 500 by Joerg Arndt, Mar 20 2016

A014138 Partial sums of (Catalan numbers starting 1, 2, 5, ...).

Original entry on oeis.org

0, 1, 3, 8, 22, 64, 196, 625, 2055, 6917, 23713, 82499, 290511, 1033411, 3707851, 13402696, 48760366, 178405156, 656043856, 2423307046, 8987427466, 33453694486, 124936258126, 467995871776, 1757900019100

Offset: 0

N. J. A. Sloane

Number of paths starting from the root in all ordered trees with n+1 edges (a path is a nonempty tree with no vertices of outdegree greater than 1). Example: a(2)=8 because the five trees with three edges have altogether 1+0+2+2+3=8 paths hanging from the roots. - Emeric Deutsch, Oct 20 2002
a(n) is the sum of the mean maximal pyramid size over all Dyck (n+1)-paths. Also, a(n) = sum of the mean maximal sawtooth size over all Dyck (n+1)-paths. A pyramid (resp. sawtooth) in a Dyck path is a subpath of the form U^k D^k (resp. (UD)^k) with k>=1 and k is its size. For example, the maximal pyramids in the Dyck path uUUDD|UD|UDdUUDD are indicated by uppercase letters (and separated by a vertical bar). Their sizes are 2,1,1,2 left to right and the mean maximal pyramid size of the path is 6/4 = 3/2. Also, the mean maximal sawtooth size of this path is (1+2+1)/3 = 4/3. - David Callan, Jun 07 2006
p^2 divides a(p-1) for prime p of form p=6k+1 (A002476(k)). - Alexander Adamchuk, Jul 03 2006
p^2 divides a(p^2-1) for prime p>3. p^2 divides a(p^3-1) for prime p=7,13,19,... prime p in the form p=6k+1. - Alexander Adamchuk, Jul 03 2006
Row sums of triangle A137614. - Gary W. Adamson, Jan 30 2008
Equals INVERTi transform of A095930: (1, 4, 15, 57, 220, 859, ...). - Gary W. Adamson, May 15 2009
a(n) < A000108(n+1), therefore A176137(n) <= 1. - Reinhard Zumkeller, Apr 10 2010
a(n) is also the sum of the numbers in Catalan's triangle (A009766) from row 0 to row n. - Patrick Labarque, Jul 27 2010
Equals the Catalan sequence starting (1, 1, 2, ...) convolved with A014137 starting (1, 2, 4, 9, ...). - Gary W. Adamson, May 20 2013
p divides a((p-3)/2) for primes {11,23,47,59,...} = A068231 primes congruent to 11 mod 12. - Alexander Adamchuk, Dec 27 2013
a(n) is the number of parking functions of size n avoiding the patterns 132, 213, and 231. - Lara Pudwell, Apr 10 2023

G. C. Greubel, Table of n, a(n) for n = 0..1000(terms 0 to 200 computed by T. D. Noe)
Ayomikun Adeniran and Lara Pudwell, Pattern avoidance in parking functions, Enumer. Comb. Appl. 3:3 (2023), Article S2R17.
Paul Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv preprint arXiv:1107.5490 [math.CO], 2011.
S. B. Ekhad and M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017)
Ângela Mestre and José Agapito, A Family of Riordan Group Automorphisms, J. Int. Seq., Vol. 22 (2019), Article 19.8.5.
Kevin Topley, Computationally Efficient Bounds for the Sum of Catalan Numbers, arXiv:1601.04223 [math.CO], 2016.

Cf. A000108, A002476, A005554, A068231, A095930, A137614, A155587.

a014138 n = a014138_list !! n
a014138_list = scanl1 (+) a000108_list  -- Reinhard Zumkeller, Mar 01 2013

a:=n->sum((binomial(2*j,j)/(j+1)),j=1..n): seq(a(n), n=0..24); # Zerinvary Lajos, Dec 01 2006
# Second program:
A014138 := series(exp(2*x)*(BesselI(0, 2*x)/2 - BesselI(1, 2*x)) + exp(x)*(3/2*int(BesselI(0, 2*x)*exp(x), x) - 1/2), x = 0, 26):
seq(n!*coeff(A014138, x, n), n = 0 .. 24); # Mélika Tebni, Aug 31 2024

Table[Sum[(2k)!/k!/(k+1)!,{k,1,n}],{n,1,70}] (* Alexander Adamchuk, Jul 03 2006 *)
Join[{0},Accumulate[CatalanNumber[Range[30]]]] (* Harvey P. Dale, Jan 25 2013 *)
CoefficientList[Series[(1 - 2 x - (1 - 4 x)^(1/2))/(2 x (1 - x)), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 21 2015 *)
a[0] := 0; a[n_] := Sum[CatalanNumber[k], {k, 1, n}]; Table[a[n], {n,0,50}] (* G. C. Greubel, Jan 14 2017 *)

Vec((1-2*x-(1-4*x)^(1/2))/(2*x*(1-x))) \\ Charles R Greathouse IV, Feb 11 2011

from _future_ import division
A014138_list, b, s = [0], 1, 0
for n in range(1,10**2):
    s += b
    A014138_list.append(s)
    b = b*(4*n+2)//(n+2) # Chai Wah Wu, Jan 28 2016

a(n) = A014137(n)-1.
G.f.: (1-2*x-sqrt(1-4x))/(2x(1-x)) = (C(x)-1)/(1-x) where C(x) is the generating function for the Catalan numbers. - Rocio Blanco, Apr 02 2007
a(n) = Sum_{k=1..n} A000108(k). - Alexander Adamchuk, Jul 03 2006
Binomial transform of A005554: (1, 2, 3, 6, 13, 30, 72, ...). - Gary W. Adamson, Nov 23 2007
D-finite with recurrence: (n+1)*a(n) + (1-5n)*a(n-1) + 2*(2n-1)*a(n-2) = 0. - R. J. Mathar, Dec 14 2011
Equals the Catalan sequence starting (1, 1, 2, ...) convolved with A014137 starting (1, 2, 4, 9, ...). - Gary W. Adamson, May 20 2013
G.f.: 1/x - G(0)/(1-x)/x, where G(k) = 1 - x/(1 - x/(1 - x/(1 - x/G(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Jul 17 2013
G.f.: 1/x - T(0)/(2*x*(1-x)), where T(k) = 2*x*(2*k+1)+ k+2 - 2*x*(k+2)*(2*k+3)/T(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 27 2013
a(n) ~ 2^(2*n+2)/(3*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Dec 10 2013
a(n) = Sum_{i+jA000108. - Yuchun Ji, Jan 10 2019
E.g.f.: exp(2*x)*(BesselI(0, 2*x)/2 - BesselI(1, 2*x)) + exp(x)*(3/2*Integral_{x=-oo..oo} BesselI(0,2*x)*exp(x) dx - 1/2). - Mélika Tebni, Aug 31 2024

Edited by Max Alekseyev, Sep 13 2009 (including adding an initial 0)

Definition edited by N. J. A. Sloane, Oct 03 2009

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).

A141123 Primes of the form -x^2+2xy+2y^2 (as well as of the form 3x^2+6xy+2*y^2).

Original entry on oeis.org

2, 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 (oscfalgan(AT)yahoo.es), Jun 05 2008

nonn

Discriminant = 12. 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.

This is exactly {2} U A068231, primes congruent to 11 (mod 12). This is because the orders of imaginary quadratic fields with discriminant 12 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) = 11 because we can write 11 = -1^2 + 2*1*2 + 2*2^2 (or 11 = 3*1^2 + 6*1*1 + 2*1^2).

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

Robert Israel, Table of n, a(n) for n = 1..6514
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 A141187.
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
Cf. A084917.

N:= 2000:
S:= NULL:
for xx from 1 to floor(2*sqrt(N/3)) do
  for yy from ceil(sqrt(max(1,3*xx^2-N))) to floor(sqrt(3)*xx) do
     S:= S, 3*xx^2-yy^2;
od od:
sort(convert(select(isprime,{S}),list)); # Robert Israel, Jul 20 2020

Reap[For[p = 2, p < 2000, p = NextPrime[p], If[FindInstance[p == -x^2 + 2*x*y + 2*y^2, {x, y}, Integers, 1] =!= {}, Print[p]; Sow[p]]]][[2, 1]]
(* or: *)
Select[Prime[Range[200]], # == 2 || # == 3 || Mod[#, 12] == 11&] (* Jean-François Alcover, Oct 25 2016, updated Oct 29 2016 *)

More terms from Colin Barker, Apr 05 2015

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

A132232 Primes congruent to 11 (mod 30).

Original entry on oeis.org

11, 41, 71, 101, 131, 191, 251, 281, 311, 401, 431, 461, 491, 521, 641, 701, 761, 821, 881, 911, 941, 971, 1031, 1061, 1091, 1151, 1181, 1301, 1361, 1451, 1481, 1511, 1571, 1601, 1721, 1811, 1871, 1901, 1931, 2081, 2111, 2141, 2351, 2381, 2411, 2441, 2531

Offset: 1

Omar E. Pol, Aug 15 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
C. K. Caldwell, The Prime Pages.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000040, A068231, A039949.

Cf. A132230, A132231, A132233, A132234, A132235, A132236.

[p: p in PrimesUpTo(3000) | p mod 30 eq 11 ]; // Vincenzo Librandi, Aug 14 2012

Select[Prime[Range[1000]],MemberQ[{11},Mod[#,30]]&] (* Vincenzo Librandi, Aug 14 2012 *)
Select[Range[11,2600,30],PrimeQ] (* Harvey P. Dale, Mar 07 2020 *)

is(n)=isprime(n) && n%30==11 \\ Charles R Greathouse IV, Jul 01 2016

From Ray Chandler, Apr 07 2009: (Start)
a(n) = A158614(n)*30 + 11.
Intersection of A030430 and A007528. (End)

Extended by Ray Chandler, Apr 07 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

A385220 Primes p such that multiplicative order of 3 modulo p is odd.

Original entry on oeis.org

2, 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, 719, 733, 743, 757, 827, 829, 839, 863, 887, 911, 947, 971, 983

Offset: 1

Jianing Song, Jun 22 2025

The multiplicative order of 3 modulo a(n) is A385226(n).
Without 2, contained in primes congruent to 1 or 11 modulo 12 (primes p such that 3 is a quadratic residue modulo p; A097933), and contains primes congruent to 11 modulo 12 (A068231).
Conjecture: this sequence has density 1/3 among the primes.

Jianing Song, Table of n, a(n) for n = 1..10000

A068231 < this sequence < A045317 < A040101 < A097933 (ignoring terms 2, 3), where Ax < Ay means that Ax is a subsequence of Ay.
Complement of A301916 in {primes} \ {3}.
Cf. A385226 (the actual multiplicative orders).
Cf. other bases: A014663 (base 2), this sequence (base 3), A385221 (base 4), A385192 (base 5), A163183 (base -2), A385223 (base -3), A385224 (base -4), A385225 (base -5).

Select[Prime[Range[200]], OddQ[MultiplicativeOrder[3, #]] &] (* Paolo Xausa, Jun 28 2025 *)

isA385220(p) = isprime(p) && (p!=3) && znorder(Mod(3,p))%2

cp's OEIS Frontend

A270596 Primes p congruent to 11 mod 12 (A068231), such that there exists a nonzero element c of GF(p), such that the element c, c-1 and -1 generate a proper subgroup of the multiplicative group.

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A014138 Partial sums of (Catalan numbers starting 1, 2, 5, ...).

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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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A141123 Primes of the form -x^2+2*x*y+2*y^2 (as well as of the form 3*x^2+6*x*y+2*y^2).

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A068229 Primes congruent to 7 (mod 12).

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A141123 Primes of the form -x^2+2xy+2y^2 (as well as of the form 3x^2+6xy+2*y^2).