A059940 -id:A059940 - cp's OEIS frontend

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

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.

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.

A060914 Integers i > 1 for which there are two primes p such that i is a solution mod p of x^3 = 2.

Original entry on oeis.org

7, 16, 20, 21, 26, 32, 34, 45, 49, 50, 52, 54, 57, 58, 61, 70, 72, 79, 81, 86, 92, 94, 98, 103, 111, 112, 114, 116, 119, 122, 125, 130, 136, 137, 141, 143, 147, 152, 157, 160, 170, 176, 179, 181, 184, 186, 197, 198, 199, 214, 221, 222, 225, 231, 234, 236, 240

Offset: 1

Klaus Brockhaus, Apr 08 2001

nonn

Solutions mod p are represented by integers from 0 to p - 1. The following equivalences holds for i > 1: There is a prime p such that i is a solution mod p of x^3 = 2 iff i^3 - 2 has a prime factor > i; i is a solution mod p of x^3 = 2 iff p is a prime factor of i^3 - 2 and p > i. i^3 - 2 has at most two prime factors > i. For i such that i^3 - 2 has no prime factors > i; cf. A060591.

			a(3) = 20, since 20 is (after 7 and 16) the third integer i for which there are two primes p > i (viz. 31 and 43) such that i is a solution mod p of x^3 = 2, or equivalently, 20^3 - 2 = 7998 = 2*3*31*43 has two prime factors > 20. (cf. A059940).

A040028, A059940, A060591.

a(n) = n-th integer i such that i^3 - 2 has two prime factors > i.

A060591 Integers i > 1 for which there is no prime p such that i is a solution mod p of x^3 = 2.

Original entry on oeis.org

113, 128, 194, 283, 333, 338, 376, 403, 430, 450, 491, 503, 548, 578, 722, 866, 875, 906, 1008, 1102, 1243, 1244, 1256, 1260, 1365, 1368, 1371, 1392, 1453, 1478, 1529, 1537, 1675, 1718, 1802, 1805, 1911, 1926, 1971, 2051, 2084, 2108, 2132, 2153, 2163

Offset: 1

Klaus Brockhaus, Apr 06 2001

nonn

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

			a(1) = 113, since there is no prime p such that 113 is a solution mod p of x^3 = 2 and for each integer i from 2 to 112 there is a prime q such that i is a solution mod q of x^3 = 2 (cf. A059940).

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

Cf. A040028, A059940.

filter:= proc(i) max(numtheory:-factorset(i^3-2)) <= i end proc:
select(filter, [$2..10000]); # Robert Israel, Apr 26 2024

Integer i > 1 is a term of this sequence iff i^3-2 has no prime factor > i.

cp's OEIS Frontend

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

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A060914 Integers i > 1 for which there are two primes p such that i is a solution mod p of x^3 = 2.

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A060591 Integers i > 1 for which there is no prime p such that i is a solution mod p of x^3 = 2.

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