A060121 -id:A060121 - cp's OEIS frontend

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

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

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.

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

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

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

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