A060124 -id:A060124 - cp's OEIS frontend

A014752 Primes of the form x^2 + 27y^2.

31, 43, 109, 127, 157, 223, 229, 277, 283, 307, 397, 433, 439, 457, 499, 601, 643, 691, 727, 733, 739, 811, 919, 997, 1021, 1051, 1069, 1093, 1327, 1399, 1423, 1459, 1471, 1579, 1597, 1627, 1657, 1699, 1723, 1753, 1777, 1789, 1801, 1831, 1933, 1999, 2017

Offset: 1

Klaus Brockhaus, Mar 02 2001

nonn

Primes p == 1 (mod 3) such that 2 is a cubic residue mod p.
Primes p == 1 (mod 6) such that 2 and -2 are both cubes (one implies the other) mod p. - Warren D. Smith
Subsequence of A040028, complement of A045309 relative to A040028. For p in this sequence, x^3 == 2 (mod p) has three solutions in integers from 0 to p-1, whose sum is p (A059899) or 2*p (A059914). The solutions are given in A060122, A060123 and A060124. - Klaus Brockhaus, Mar 02 2001
Primes p = 3m+1 such that 2^m == 1 (mod p). Subsequence of A016108 which also includes composites satisfying this congruence. - Alzhekeyev Ascar M, Feb 22 2012

K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer, 1982, Prop. 9.6.2, p. 119.

N. J. A. Sloane and T. D. Noe, Table of n, a(n) for n = 1..17753 (The first 1000 terms were computed by T. D. Noe)
Steven R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
Bishnu Paudel and Chris Pinner, The integer group determinants for the abelian groups of order 18, arXiv:2412.10638 [math.NT], 2024. See p. 3.
Zak Seidov, Corresponding values of x and y
Bram van Asch, On the structure of the ring Z[2^(1/3)], Internat. J. Pure Appl. Math., 16 (No. 2, 2004), 243-251. See Prop. 7.

Cf. A040028, A045309, A059899, A059914, A060122, A060123, A060124, A014753.

[p: p in PrimesUpTo(2500) | NormEquation(27, p) eq true]; // Vincenzo Librandi, Jul 24 2016

With[{nn=50},Take[Select[Union[First[#]^2+27Last[#]^2&/@Tuples[Range[ nn], 2]],PrimeQ],nn]] (* Harvey P. Dale, Jul 28 2014 *)
nn = 1398781;re = Sort[Reap[Do[Do[If[PrimeQ[p = x^2 + 27*y^2], Sow[{p, x, y}]], {x, Sqrt[nn - 27*y^2]}], {y, Sqrt[nn/27]}]][[2, 1]]]; (* For all 17753 values of a(n), x(n) and y(n). - Zak Seidov, May 20 2016 *)

{ fc(a,b,c,M) = my(p,t1,t2,n); t1 = listcreate();
for(n=1,M, p = prime(n);
t2 = qfbsolve(Qfb(a,b,c),p); if(t2 == 0,, listput(t1,p)));
print(t1);
}
fc(1,0,27,1000);
\\ N. J. A. Sloane, Jun 06 2014

list(lim)=my(v=List()); forprimestep(p=31,lim,6, if(Mod(2,p)^(p\3)==1, listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Apr 06 2022

a(n) ~ 6n log n by the Landau prime ideal theorem. - Charles R Greathouse IV, Apr 06 2022

Definition provided by T. D. Noe, May 08 2005
Entry revised by Michael Somos and N. J. A. Sloane, Jul 28 2006
Defective Mma program replaced with PARI program, b-file recomputed and extended by N. J. A. Sloane, Jun 06 2014

A059940 Smallest prime p such that x = n is a solution mod p of x^3 = 2, or 0 if no such prime exists.

Original entry on oeis.org

3, 5, 31, 41, 107, 11, 17, 727, 499, 443, 863, 439, 457, 3373, 23, 1637, 53, 6857, 31, 47, 5323, 811, 6911, 919, 29, 19681, 439, 739, 13499, 29789, 43, 7187, 43, 461, 23327, 50651, 59, 2579, 2909, 22973, 2179, 15901, 14197, 293, 1187, 34607, 11059

Offset: 2

Klaus Brockhaus, Mar 02 2001

nonn

Solutions mod p are represented by integers from 0 to p-1. The following equivalences hold for n > 1: There is a prime p such that n is a solution mod p of x^3 = 2 iff n^3-2 has a prime factor > n; n is a solution mod p of x^3 = 2 iff p is a prime factor of n^3-2 and p > n.

n^3-2 has at most two prime factors > n, consequently these factors are the only primes p such that n is a solution mod p of x^3 = 2. For n such that n^3-2 has no prime factor > n (the zeros in the sequence; they occur beyond the last entry shown in the database) see A060591. For n such that n^3-2 has two prime factors > n, cf. A060914.

			a(2) = 3, since 2 is a solution mod 3 of x^3 = 2 and 2 is not a solution mod p of x^3 = 2 for prime p = 2. Although 2^3 = 2 mod 2, prime 2 is excluded because 0 < 2 and 2 = 0 mod 2. a(5) = 41, since 5 is a solution mod 41 of x^3 = 2 and 5 is not a solution mod p of x^3 = 2 for primes p < 41. Although 5^3 = 2 mod 3, prime 3 is excluded because 3 < 5 and 5 = 2 mod 3.

Cf. A040028, A060121, A060122, A060123, A060124, A060591, A060914.

If n^3-2 has prime factors > n, then a(n) = least of these prime factors, else a(n) = 0.

A060121 First solution mod p of x^3 = 2 for primes p such that only one solution exists.

Original entry on oeis.org

0, 2, 3, 7, 8, 16, 26, 5, 21, 18, 38, 49, 50, 16, 26, 6, 81, 54, 98, 70, 157, 161, 58, 147, 21, 86, 92, 197, 50, 249, 137, 184, 119, 45, 45, 261, 198, 61, 176, 143, 51, 103, 221, 72, 11, 219, 35, 86, 385, 384, 141, 143, 225, 92, 245, 533, 557, 473, 170, 375, 516

Offset: 1

Klaus Brockhaus, Mar 02 2001

nonn

Solutions mod p are represented by integers from 0 to p-1. For i > 1, i is a solution mod p of x^3 = 2 iff p is a prime factor of i^3-2 and p > i (cf. comment to A059940). i^3-2 has at most two prime factors > i. Hence i is a solution mod p of x^3 = 2 for at most two different p and therefore no integer occurs more than twice in this sequence. There are integers which do occur twice, e.g. 16, 21, 26 (cf. A060914). Moreover, no integer occurs more than twice in A060121, A060122, A060123 and A060124 taken together.

			a(9) = 21, since 47 is the ninth term of A045309 and 21 is the only solution mod 47 of x^3 = 2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A040028, A045309, A059940, A060914, A060122, A060123, A060124.

Res:=0,2: count:= 2: p:= 3:
while count < 100 do
p:= nextprime(p);
   if p mod 3 = 2 then
    count:= count+1;
    Res:= Res, numtheory:-mroot(2,3,p);
fi
od:
Res; # Robert Israel, Sep 12 2018

terms = 100;
A045309 = Select[Prime[Range[2 terms]], Mod[#, 3] != 1&];
a[n_] := PowerMod[2, 1/3, A045309[[n]]];
Array[a, terms] (* Jean-François Alcover, Feb 27 2019 *)

a(n) = first (only) solution mod p of x^3 = 2, where p is the n-th prime such that x^3 = 2 has only one solution mod p, i.e. p is the n-th term of A045309.

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.

cp's OEIS Frontend

A014752 Primes of the form x^2 + 27y^2.

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A059940 Smallest prime p such that x = n is a solution mod p of x^3 = 2, or 0 if no such prime exists.

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A060121 First solution mod p of x^3 = 2 for primes p such that only one solution exists.

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

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

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