A040028 -id:A040028 - cp's OEIS frontend

A065906 Integers i > 1 for which there are three primes p such that i is a solution mod p of x^4 = 2.

15, 48, 55, 197, 206, 221, 235, 283, 297, 408, 444, 472, 489, 577, 578, 623, 641, 677, 701, 703, 763, 854, 930, 1049, 1081, 1134, 1140, 1159, 1160, 1201, 1253, 1303, 1311, 1328, 1374, 1385, 1415, 1458, 1459, 1495, 1501, 1517, 1557, 1585, 1714, 1723, 1726

Offset: 1

Klaus Brockhaus, Nov 28 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^4 = 2 iff i^4 - 2 has a prime factor > i; i is a solution mod p of x^4 = 2 iff p is a prime factor of i^4 - 2 and p > i. i^4 - 2 has at most three prime factors > i. For i such that i^4 - 2 has no resp. one resp. two prime factors > i cf. A065903 resp. A065904 resp. A065905.

			a(3) = 55, since 55 is (after 15 and 48) the third integer i for which there are three primes p > i (viz. 73, 103 and 1217) such that i is a solution mod p of x^4 = 2, or equivalently, 55^4 - 2 = 9150623 = 73*103*1217 has three prime factors > 4. (cf. A065902).

Cf. A040028, A065902, A065903, A065904, A065905.

a065906(m) = local(c,n,f,a,s,j); c = 0; n = 2; while(cn,s = concat(s,f[j,1]))); if(matsize(s)[2] == 3,print1(n,","); c++); n++)
a065906(50)

a(n) = n-th integer i such that i^4 - 2 has three prime factors > i.

A057760 Numbers n such that 2 is a cube mod n.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 11, 15, 17, 22, 23, 25, 29, 30, 31, 33, 34, 41, 43, 46, 47, 50, 51, 53, 55, 58, 59, 62, 66, 69, 71, 75, 82, 83, 85, 86, 87, 89, 93, 94, 101, 102, 106, 107, 109, 110, 113, 115, 118, 121, 123, 125, 127, 129, 131, 137, 138, 141, 142

Offset: 1

N. J. A. Sloane, Nov 01 2000

nonn

Numbers not divisible by 4 or 9 and whose prime divisors are in A040028. - Eric M. Schmidt, Jan 25 2014

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

Cf. A057126, A057761.

with(numtheory); [seq(mroot(2,3,p),p=1..400)];

is(n)=ispower(Mod(2,n),3) \\ Charles R Greathouse IV, Apr 05 2012

Checked by T. D. Noe, Apr 19 2007

A059899 Primes p such that x^3 = 2 has more than one solution mod p and the sum of the (three) solutions is p.

Original entry on oeis.org

31, 229, 397, 439, 457, 499, 601, 643, 691, 727, 739, 811, 919, 997, 1021, 1051, 1093, 1327, 1459, 1657, 1699, 1753, 1933, 1999, 2113, 2179, 2203, 2251, 2281, 2341, 2347, 2383, 2671, 2731, 2767, 2791, 2833, 2953, 2971, 3061, 3229, 3259, 3331, 3373, 3391

Offset: 1

Klaus Brockhaus, Mar 02 2001

nonn

Subsequence of A040028 and of A014752, complement of A059914 relative to A014752. Solutions mod p are represented by integers from 0 to p-1.

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

Cf. A040028, A014752, A059914.

filter:= proc(p) local S;
  if not isprime(p) then return false fi;
  S:= map(t -> rhs(t[1]), [msolve(x^3=2,p)]);
  nops(S) = 3 and convert(S,`+`) = p
end proc:
select(filter, [seq(i,i=7..5000, 6)]); # Robert Israel, Aug 13 2024

A059914 Primes p such that x^3 = 2 has more than one solution mod p and the sum of the (three) solutions is 2*p.

Original entry on oeis.org

43, 109, 127, 157, 223, 277, 283, 307, 433, 733, 1069, 1399, 1423, 1471, 1579, 1597, 1627, 1723, 1777, 1789, 1801, 1831, 2017, 2089, 2143, 2287, 2689, 2749, 2917, 3163, 3181, 3271, 3343, 3541, 3607, 3631, 3823, 3889, 4057, 4129, 4153, 4177, 4339, 4513

Offset: 1

Klaus Brockhaus, Mar 02 2001

nonn

Subsequence of A040028 and of A014752, complement of A059899 relative to A014752. Solutions mod p are represented by integers from 0 to p-1.

Cf. A040028, A014752, A059899, A001133.

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.

A070180 Primes p such that x^3 = 2 has a solution mod p, but x^(3^2) = 2 has no solution mod p.

Original entry on oeis.org

109, 307, 433, 739, 811, 919, 1423, 1459, 1999, 2017, 2143, 2179, 2251, 2287, 2341, 2791, 2917, 2953, 3061, 3259, 3331, 3457, 3889, 4177, 4339, 4519, 4663, 5113, 5167, 5419, 5437, 5653, 6301, 6427, 6661, 6679, 6967, 7723, 7741, 8011, 8389, 8713

Offset: 1

Klaus Brockhaus, Apr 29 2002

Cf. A040028, A049596, A059262, A059667, A070179, A070181 - A070188.

[p: p in PrimesUpTo(10000) | not exists{x: x in ResidueClassRing(p) | x^9 eq 2} and exists{x: x in ResidueClassRing(p) | x^3 eq 2}]; // Vincenzo Librandi, Sep 21 2012

PARI
```
forprime(p=2,8800,x=0; while(x
				
```

ok(p, r, k1, k2)={
    if (  Mod(r,p)^((p-1)/gcd(k1,p-1))!=1, return(0) );
    if (  Mod(r,p)^((p-1)/gcd(k2,p-1))==1, return(0) );
    return(1);
}
forprime(p=2,10^4, if (ok(p,2,3,3^2),print1(p,", ")));
/* Joerg Arndt, Sep 21 2012 */

A097142 Number of primes p < 10^n for which 2 is a cubic residue (mod p).

Original entry on oeis.org

3, 16, 112, 818, 6367, 52299, 442972, 3840740, 33898001, 303369367, 2745366812, 25071938615

Offset: 1

Robert G. Wilson v, Jul 26 2004

It would appear that about two-thirds of the primes have 2 as a cubic residue. A097142/A006880.

Eric Weisstein's World of Mathematics, Cubic Residue

Cf. A006880, A040028.

f[p_] := Block[{k = 2}, While[k < p && Mod[k^3, p] != 2, k++ ]; If[k == p, 0, 1]]; c = 1; k = 2; Do[ While[ p = Prime[k]; p < 10^n, If[ f[p] == 1, c++ ]; k++ ]; Print[c], {n, 5}]

a(6)-a(7) from Hiroaki Yamanouchi, Aug 31 2014

a(8)-a(12) from Hiroaki Yamanouchi, Oct 17 2015

A259732 Numbers n > 1 that divide ((p-1)/2)^3 + 2 for some odd prime p.

Original entry on oeis.org

2, 3, 5, 10, 11, 17, 22, 23, 29, 31, 34, 41, 43, 46, 47, 53, 58, 59, 62, 71, 82, 83, 86, 89, 94, 101, 106, 107, 109, 113, 118, 121, 127, 131, 137, 142, 149, 157, 166, 167, 173, 178, 179, 187, 191, 197, 202

Offset: 1

Miguel Angel Velilla Mula, Jul 04 2015

nonn

n = 3,5,10 works only once, for p=3 (3-1)/2=1, then 1^3 + 2 = 3 and for p=5 (5-1)/2=2, then 2^3+2 = 10.
This sequence is a subset of A057760, where all elements that are multiples of 3 and 5 are excluded, except the three above (3,5,10).
"Mirror sequence" of this one, when n divides ((p+1)/2)^3 - 2, p = prime, produces a sequence very close to this one, the only differences being 10 (excluded), 25 (included for p=5 (p+1)/2=3 then 3^3-2 = 25) and 6 (included for p=3 (p+1)/2=2 then 2^3-2 = 6).
Analyzing ((p-1)/2)^3 + 2 = (p^3 - 3(p(p-1)-5))/8, every composite x (mod 3) trying to divide this one will fail.
To prove 5 can't divide ((p-1)/2)^3 + 2 = (p^3 - 3p^2 + 3p + 15)/8 we use the last digit of p, which can be 1,3,7 or 9. This leads the last digit of the formula to be (1,9,7 or 3) + 15, so it cannot be divided by 5, unless the last digit of p is 5. This happens just for the only prime divisible by 5, i.e., 5 itself, which occurs only once.
A179871 looks very similar to this sequence.

Cf. A057760, A040028.

Select[Rest[Union[Flatten[Divisors/@Rest[Table[((p-1)/2)^3+2,{p,Prime[Range[2000]]}]]]]],#<250&] (* Harvey P. Dale, Jun 01 2025 *)

cp's OEIS Frontend

A065906 Integers i > 1 for which there are three primes p such that i is a solution mod p of x^4 = 2.

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A057760 Numbers n such that 2 is a cube mod n.

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A059899 Primes p such that x^3 = 2 has more than one solution mod p and the sum of the (three) solutions is p.

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A059914 Primes p such that x^3 = 2 has more than one solution mod p and the sum of the (three) solutions is 2*p.

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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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A070180 Primes p such that x^3 = 2 has a solution mod p, but x^(3^2) = 2 has no solution mod p.

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Magma

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PARI

A097142 Number of primes p < 10^n for which 2 is a cubic residue (mod p).

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Mathematica

Extensions

A259732 Numbers n > 1 that divide ((p-1)/2)^3 + 2 for some odd prime p.

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Mathematica