A014752 -id:A014752 - cp's OEIS frontend

A059898 Duplicate of A014752.

31, 43, 109, 127, 157, 223, 229, 277, 283, 307, 397, 433, 439, 457, 499, 601, 643, 691

Offset: 1

dead

A107132 Primes of the form 2x^2 + 13y^2.

2, 13, 31, 149, 167, 317, 359, 397, 463, 487, 509, 613, 661, 709, 839, 1061, 1087, 1103, 1151, 1181, 1367, 1471, 1783, 1789, 1861, 2039, 2111, 2221, 2269, 2437, 2503, 2621, 2647, 2917, 2927, 2957, 3023, 3079, 3167, 3229, 3373, 3541, 3853

Offset: 1

T. D. Noe, May 13 2005

Discriminant = -104. Binary quadratic forms ax^2+cy^2 have discriminant d=-4ac. We consider sequences of primes produced by forms with -400<=d<=0, a<=c and gcd(a,c)=1. These restrictions yield 173 sequences of prime numbers, which are organized by discriminant below. See A106856 for primes of the form ax^2+bxy+cy^2 with discriminant > -100.

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.

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A033218 (d=-104), A014752 (d=-108), A107133, A107134 (d=-112), A033219 (d=-116), A107135-A107137, A033220 (d=-120), A033221 (d=-124), A105389 (d=-128), A107138, A033222 (d=-132), A107139, A033223 (d=-136), A107140, A033224 (d=-140), A107141, A107142 (d=-144), A033225 (d=-148), A107143, A033226 (d=-152), A033227 (d=-156), A107144, A107145 (d=-160), A033228 (d=-164), A107146-A107148, A033229 (d=-168).
Cf. A033230 (d=-172), A107149, A107150 (d=-176), A107151, A107152 (d=-180), A107153, A033231 (d=-184), A033232 (d=-188), A141373 (d=-192), A107155 (d=-196), A107156, A107157 (d=-200), A107158, A033233 (d=-204), A107159, A107160 (d=-208), A033234 (d=-212), A107161, A107162 (d=-216), A033235 (d=-220), A107163, A107164 (d=-224), A107165, A033236 (d=-228), A107166, A033237 (d=-232), A033238 (d=-236).
Cf. A107167-A107169 (d=-240), A033239 (d=-244), A107170, A033240 (d=-248), A014754 (d=-256), A107171, A033241 (d=-260), A107172-A107174, A033242 (d=-264), A033243 (d=-268), A107175, A107176 (d=-272), A107177, A033244 (d=-276), A107178-A107180, A033245 (d=-280), A033246 (d=-284), A107181 (d=-288), A033247 (d=-292), A107182, A033248 (d=-296), A107183, A107184 (d=-300), A107185, A107186 (d=-304), A107187, A033249 (d=-308).
Cf. A107188-A107190, A033250 (d=-312), A033251 (d=-316), A107191, A107192 (d=-320), A107193 (d=-324), A107194, A033252 (d=-328), A033253 (d=-332), A107195-A107198 (d=-336), A107199, A033254 (d=-340), A107200, A033255 (d=-344), A033256 (d=-348), A107132 A107201, A107202 (d=-352), A033257 (d=-356), A107203-A107206 (d=-360), A107207, A033258 (d=-364), A107208, A107209 (d=-368), A107210, A033202 (d=-372).
Cf. A107211, A033204 (d=-376), A033206 (d=-380), A107212, A107213 (d=-384), A033208 (d=-388), A107214, A107215 (d=-392), A107216, A107217 (d=-396), A107218, A107219 (d=-400).
For a more complete list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

QuadPrimes2[2, 0, 13, 10000] (* see A106856 *)

list(lim)=my(v=List([2,13]),t); for(y=1,sqrtint(lim\13), for(x=1,sqrtint((lim-13*y^2)\2), if(isprime(t=2*x^2+13*y^2), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Feb 07 2017

A040028 Primes p such that x^3 = 2 has a solution mod p.

Original entry on oeis.org

2, 3, 5, 11, 17, 23, 29, 31, 41, 43, 47, 53, 59, 71, 83, 89, 101, 107, 109, 113, 127, 131, 137, 149, 157, 167, 173, 179, 191, 197, 223, 227, 229, 233, 239, 251, 257, 263, 269, 277, 281, 283, 293, 307, 311, 317, 347, 353, 359, 383, 389, 397, 401, 419, 431, 433

Offset: 1

N. J. A. Sloane

This is the union of {3}, A003627 (primes congruent to 2 mod 3) and A014752 (primes of the form x^2+27y^2). By Thm. 4.15 of [Cox], p is of the form x^2+27y^2 if and only if p is congruent to 1 mod 3 and 2 is a cubic residue mod p. If p is not congruent to 1 mod 3, then every number is a cubic residue mod p, including 2. - Andrew V. Sutherland, Apr 26 2008

Complement of A040034 relative to A000040. - Vincenzo Librandi, Sep 13 2012

David A. Cox, "Primes of the Form x^2+ny^2", 1998, John Wiley & Sons.
Kenneth Ireland and Michael Rosen, "A Classical Introduction to Modern Number Theory", second ed., 1990, Springer-Verlag.

T. D. Noe, Table of n, a(n) for n = 1..1000
Index entries for related sequences

Cf. A001132. Number of primes p < 10^n for which 2 is a cubic residue (mod p) is in A097142.
Cf. A000040, A003627, A014752, A040034.
For primes p such that x^m == 2 mod p has a solution for m = 2,3,4,5,6,7,... see A038873, A040028, A040098, A040159, A040992, A042966, ...

[ p: p in PrimesUpTo(433) | exists(t){x : x in ResidueClassRing(p) | x^3 eq 2} ]; // Klaus Brockhaus, Dec 02 2008

f[p_] := Block[{k = 2}, While[k < p && Mod[k^3, p] != 2, k++ ]; If[k == p, 0, 1]]; Select[ Prime[ Range[100]], f[ # ] == 1 &] (* Robert G. Wilson v, Jul 26 2004 *)

select(p->ispower(Mod(2,p),3),primes(100)) \\ Charles R Greathouse IV, Apr 28 2015

a(n) ~ (3/2) n log n. - Charles R Greathouse IV, Apr 06 2022

Typo corrected to A014752 by Paul Landon (paullandon(AT)hotmail.com), Jan 25 2010

A001133 Primes p such that the multiplicative order of 2 modulo p is (p-1)/3.

Original entry on oeis.org

43, 109, 157, 229, 277, 283, 307, 499, 643, 691, 733, 739, 811, 997, 1021, 1051, 1069, 1093, 1459, 1579, 1597, 1627, 1699, 1723, 1789, 1933, 2179, 2203, 2251, 2341, 2347, 2749, 2917, 3163, 3181, 3229, 3259, 3373, 4027, 4339, 4549, 4597, 4651, 4909, 5101, 5197, 5323, 5413, 5437, 5653, 6037

Offset: 1

N. J. A. Sloane

nonn

M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 59.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A040028, A014752, A059914.

Cf. also A001134, A001135, A001136, A115586, A115591.

[ p: p in PrimesUpTo(4597) | r eq 1 and Order(R!2) eq q where q,r is Quotrem(p,3) where R is ResidueClassRing(p) ]; // Klaus Brockhaus, Dec 02 2008

Reap[For[p = 2, p < 10^4, p = NextPrime[p], If[MultiplicativeOrder[2, p] == (p-1)/3, Sow[p]]]][[2, 1]] (* Jean-François Alcover, Dec 10 2015 *)

forprime(p=3,10^4,if(znorder(Mod(2,p))==(p-1)/3,print1(p,", "))); \\ Joerg Arndt, May 17 2013

More terms and better definition from Don Reble, Mar 11 2006

A045309 Primes congruent to {0, 2} mod 3.

Original entry on oeis.org

2, 3, 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

Offset: 1

N. J. A. Sloane

Also, primes p such that the equation x^3 == y (mod p) has a unique solution x for every choice of y. - Klaus Brockhaus, Mar 02 2001; Michel Drouzy (DrouzyM(AT)noos.fr), Oct 28 2001

2, 3 and primes congruent to 5 mod 6. - Chai Wah Wu, Apr 28 2025

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

Cf. A040028, A014752, A060121, A003627, A007528, A045410.

[ p: p in PrimesUpTo(1000) | #[ x: x in ResidueClassRing(p) | x^3 eq 2 ] eq 1 ]; // Klaus Brockhaus, Apr 11 2009

Select[Prime[Range[150]],MemberQ[{0,2},Mod[#,3]]&] (* Harvey P. Dale, Jun 14 2011 *)

is(n)=isprime(n) && n%3!=1 \\ Charles R Greathouse IV, Apr 20 2015

from itertools import count, islice
from sympy import isprime
def A045309_gen(): # generator of terms
    yield from (2,3)
    yield from filter(isprime, count(5,6))
A045309_list = list(islice(A045309_gen(),48)) # Chai Wah Wu, Apr 28 2025

a(n) ~ 2n log n. - Charles R Greathouse IV, Apr 20 2015

Edited by N. J. A. Sloane, Apr 11 2009

A002225 a(n) is the smallest prime p such that each of the first n primes has three cube roots mod p.

Original entry on oeis.org

31, 307, 643, 5113, 21787, 39199, 360007, 360007, 4775569, 10318249, 10318249, 65139031, 387453811, 913900417, 2278522747, 2741702809, 25147657981, 118748663779, 156776294593, 747206701687, 1151810360731, 1151810360731, 1151810360731

Offset: 1

N. J. A. Sloane

a(n) is the smallest prime p == 1 (mod 3) such that each of the first n primes is a cubic residue mod p. - Robert Israel, Aug 02 2016

			For n = 2, the first two primes 2 and 3 each have three cube roots mod 307: 2 has cube roots 52, 270, 292 and 3 has cube roots 192, 194, 228. - _Robert Israel_, Aug 02 2016

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. E. Western and J. C. P. Miller, Tables of Indices and Primitive Roots. Royal Society Mathematical Tables, Vol. 9, Cambridge Univ. Press, 1968, p. XVI.

A. E. Western and J. C. P. Miller, Tables of Indices and Primitive Roots, Royal Society Mathematical Tables, Vol. 9, Cambridge Univ. Press, 1968 [Annotated scans of selected pages]

Smallest prime p such that each of the first n primes has q q-th roots mod p: A147972 (q=2), this sequence (q=3), A002226 (q=5), A002227 (q=7), A002228 (q=11), A060363 (q=13), A060364 (q=17).
Cf. A002223, A002224.
Subset of A014752. Except for a(1), subset of A014753. Except for a(1) and a(2), subset of A040044.

Primes:= [2]: pp:= 7:
for n from 1 to 12 do
  for p from pp by 6 while
    not(isprime(p) and andmap(t -> t &^ ((p-1)/3) mod p = 1, Primes))
  do od:
  A[n]:= p;
  pp:= p;
  Primes:= [op(Primes), nextprime(Primes[-1])];
od:
seq(A[i],i=1..12); # Robert Israel, Aug 02 2016

(* This naive program being very slow, limit is set to 8 terms *) lim=8; np[] := While[p=NextPrime[p]; Mod[p,3]!=1]; crQ[n_, p_] := Reduce[ 0A002225={}; While[Length[A002225] < lim, If[And @@ (crQ[#,p]& /@ pp), AppendTo[pp, NextPrime[ Last[pp]]]; Print[p]; AppendTo[A002225,p], np[] ] ]; A002225 (* Jean-François Alcover, Sep 09 2011 *)

More terms from Don Reble, Oct 09 2001
Name corrected by Robert Israel, Aug 02 2016
a(18)-a(23) from Sergey Paramonov, Apr 11 2024

A014753 Primes p==1 (mod 6) such that 3 and -3 are both cubes (one implies other) modulo p.

Original entry on oeis.org

61, 67, 73, 103, 151, 193, 271, 307, 367, 439, 499, 523, 547, 577, 613, 619, 643, 661, 727, 757, 787, 853, 919, 967, 991, 997, 1009, 1021, 1093, 1117, 1249, 1303, 1321, 1399, 1531, 1543, 1549, 1597, 1609, 1621, 1669, 1759, 1783, 1861, 1867

Offset: 1

Warren D. Smith

Primes of the form x^2+xy+61y^2, whose discriminant is -243. - T. D. Noe, May 17 2005

Primes of the form (x^2 + 243*y^2)/4. - Arkadiusz Wesolowski, May 30 2015

K. Ireland and M. Rosen, A classical introduction to modern number theory, Vol. 84, Graduate Texts in Mathematics, Springer-Verlag. Exercise 23, p. 135.

Bruno Berselli, Table of n, a(n) for n = 1..1000

Cf. A014752, A014755.

p6 = Select[6*Range[0, 400]+1, PrimeQ]; Select[p6, (Reduce[3 == k^3+m*#, {k, m}, Integers] =!= False)&] (* Jean-François Alcover, Feb 20 2014 *)

forprime(p=1, 9999, p%6==1&&ispower(Mod(3, p), 3)&&print1(p", ")) \\ M. F. Hasler, Feb 18 2014

is_A014753(p)={p%6==1&&ispower(Mod(3, p), 3)&&isprime(p)} \\ M. F. Hasler, Feb 18 2014

Offset changed from 0 to 1 by Bruno Berselli, Feb 20 2014

A060122 Smallest solution mod p of x^3 = 2 for primes p such that more than one solution exists.

Original entry on oeis.org

4, 20, 57, 32, 62, 68, 52, 152, 120, 52, 53, 72, 13, 14, 10, 54, 61, 94, 9, 339, 29, 23, 25, 114, 159, 131, 469, 206, 178, 892, 628, 162, 544, 709, 647, 799, 49, 57, 709, 218, 1118, 585, 858, 332, 528, 119, 1151, 1024, 152, 798, 42, 235, 71, 535, 733, 257, 228

Offset: 1

Klaus Brockhaus, Mar 02 2001

nonn

Solutions mod p are represented by integers from 0 to p-1. No integer occurs more than twice in this sequence (cf. comment to A060121). There are integers which do occur twice, e.g. 52, 57, 152 (cf. A060914). Moreover, no integer occurs more than twice in A060121, A060122, A060123 and A060124 taken together.

			a(3) = 57, since 109 is the third term of A014752, 57, 58 and 103 are the solutions mod 109 of x^3 = 2 and 57 is the least one.

Cf. A040028, A014752, A059940, A060914, A060121, A060123, A060124.

a(n) = first (least) solution mod p of x^3 = 2, where p is the n-th prime such that x^3 = 2 has more than one solution mod p, i.e. p is the n-th term of A014752.

A060123 Second solution mod p of x^3 = 2 for primes p such that more than one solution exists.

Original entry on oeis.org

7, 32, 58, 100, 116, 179, 79, 181, 186, 270, 130, 394, 28, 34, 97, 94, 73, 288, 348, 407, 298, 231, 381, 125, 315, 458, 781, 385, 425, 928, 1095, 362, 1186, 992, 1046, 1053, 116, 542, 1236, 425, 1129, 1259, 1344, 1553, 570, 200, 1328, 1286, 888, 1433, 808

Offset: 1

Klaus Brockhaus, Mar 02 2001

nonn

Solutions mod p are represented by integers from 0 to p-1. No integer occurs more than twice in this sequence (cf. comment to A060121). There are integers which do occur twice, e.g. 116, 425 (cf. A060914). Moreover, no integer occurs more than twice in A060121, A060122, A060123 and A060124 taken together.

			a(3) = 58, since 109 is the third term of A014752 and 58 is the second solution mod 109 of x^3 = 2.

Cf. A040028, A014752, A059940, A060914, A060121, A060122, A060124.

a(n) = second solution mod p of x^3 = 2, where p is the n-th prime such that x^3 = 2 has more than one solution mod p, i.e. p is the n-th term of A014752.

A060124 Third solution mod p of x^3 = 2 for primes p such that more than one solution exists.

Original entry on oeis.org

20, 34, 103, 122, 136, 199, 98, 221, 260, 292, 214, 400, 398, 409, 392, 453, 509, 309, 370, 720, 412, 557, 513, 758, 547, 462, 888, 502, 724, 978, 1123, 935, 1212, 1457, 1501, 1402, 1492, 1100, 1501, 1110, 1307, 1734, 1400, 1777, 835, 1680, 1555, 1868

Offset: 1

Klaus Brockhaus, Mar 02 2001

nonn

Solutions mod p are represented by integers from 0 to p-1. No integer occurs more than twice in this sequence (cf. comment to A060121). There are integers which do occur twice, e.g. 1501 (cf. A060914). Moreover, no integer occurs more than twice in A060121, A060122, A060123 and A060124 taken together.

			a(3) = 103, since 109 is the third term of A014752 and 103 is the third solution mod 109 of x^3 = 2.

Cf. A040028, A014752, A059940, A060914, A060121, A060122, A060123.

a(n) = third solution mod p of x^3 = 2, where p is the n-th prime such that x^3 = 2 has more than one solution mod p, i.e. p is the n-th term of A014752.

cp's OEIS Frontend

A059898 Duplicate of A014752.

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A107132 Primes of the form 2x^2 + 13y^2.

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A040028 Primes p such that x^3 = 2 has a solution mod p.

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A001133 Primes p such that the multiplicative order of 2 modulo p is (p-1)/3.

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A045309 Primes congruent to {0, 2} mod 3.

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A002225 a(n) is the smallest prime p such that each of the first n primes has three cube roots mod p.

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A014753 Primes p==1 (mod 6) such that 3 and -3 are both cubes (one implies other) modulo p.

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