A040034 -id:A040034 - cp's OEIS frontend

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

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

A259189 Semiprimes of the form n^3 + 2.

Original entry on oeis.org

10, 218, 514, 731, 1333, 2199, 2746, 3377, 4915, 5834, 6861, 8002, 9263, 12169, 15627, 29793, 35939, 42877, 54874, 59321, 68923, 117651, 125002, 132653, 148879, 185195, 205381, 314434, 405226, 421877, 474554, 531443, 592706, 658505, 704971

Offset: 1

Morris Neene, Jun 20 2015

nonn

Intersection of A001358 and A084380. - Michel Marcus, Jun 20 2015
Since there are no squares of the form n^3 + 2, all semiprimes in this sequence are products of distinct primes.
No term in A040034 divides any term in this sequence.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

Cf. A001358 (semiprimes), A084380 (n^3+2), A144953 (primes of same form).

Cf. A237040 (similar sequence with n^3+1).

IsSP:=func;[r:n in [1..1000]|IsSP(r) where r is 2+n^3];

Select[Range[100]^3 + 2, PrimeOmega[#] == 2 &] (* Alonso del Arte, Jun 20 2015 *)

is(n)=bigomega(n^3 + 2)==2 \\ Anders Hellström, Sep 07 2015

use ntheory ":all"; my @sp = grep { scalar(factor($))==2 } map { $**3+2 } 1..100; say "@sp"; # Dana Jacobsen, Sep 07 2015

A107662 -n is the discriminant of cubic polynomials irreducible over Zp for primes p represented by only one binary quadratic form.

Original entry on oeis.org

23, 31, 44, 59, 76, 83, 107, 108, 139, 172, 211, 243, 268, 283, 307, 331, 379, 499, 547, 643, 652, 883, 907

Offset: 1

T. D. Noe, May 19 2005

Let f(x) be any monic integral cubic polynomial with discriminant -n and irreducible over Z. Consider the set S of primes p such that f(x) has no zeros in Zp, i.e., f(x) is irreducible in Zp. For the discriminants -n in this sequence, set S coincides with the primes represented by one binary quadratic form ax^2+bxy+cy^2 with -n=b^2-4ac. For examples, see A106867, A106872, A106282, A106919, A106954, A106967, A040034 and A040038. This sequence consists of (1) terms 4d in A106312 such that the class number of d is 1, (2) terms d in A106312 such that the class number of d is 3 and (3) 108 and 243.

			For each -n, we give (-n,a,b,c) for the quadratic form ax^2+bxy+cy^2: (23,2,1,3), (31,2,1,4), (44,3,2,4), (59,3,1,5), (76,4,2,5), (83,3,1,7), (107,3,1,9), (108,4,2,7), (139,5,1,7), (172,4,2,11), (211,5,3,11), (243,7,3,9), (268,4,2,17), (283,7,5,11), (307,7,1,11), (331,5,3,17), (379,5,1,19), (499,5,1,25), (547,11,5,13), (643,7,1,23), (652,4,2,41), (883,13,1,17) and (907,13,9,19).

Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251-257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012.
Blair K. Spearman and Kenneth S. Williams, The cubic congruence x^3+Ax^2+Bx+C = 0 (mod p) and binary quadratic forms, J. London Math. Soc., 46, (1992), 397-410.

Eric Weisstein's World of Mathematics, Class Number

Cf. A106312 (possible negative discriminants of cubic polynomials), A014602 (negative discriminants having class number 1), A006203 (negative discriminants having class number 3).

A162541 Primes p such that a splitting of the cyclic group Zp by the perfect 3-shift code {+-1,+-2,+-3} exists.

Original entry on oeis.org

7, 37, 139, 163, 181, 241, 313, 337, 349, 379, 409, 421, 541, 571, 607, 631, 751, 859, 877, 937, 1033, 1087, 1123, 1171, 1291, 1297, 1447, 1453, 1483, 1693, 1741, 1747, 2011, 2161, 2239, 2311, 2371, 2473, 2539, 2647, 2677, 2707, 2719, 2857, 3169, 3361, 3433, 3511, 3547

Offset: 1

Ctibor O. Zizka, Jul 05 2009

nonn

This list was computed by S. Saidi.
From Travis Scott, Oct 04 2022: (Start)
These are also the p whose (phi/3)-th power residues have minimal bases at {1,2,3} (see under Example). Such covers {1a(n)-> {1,2,3}(n) =   7,  37,  139,  163,  181,  241, ... ~    (9*n)*log(n)
       {1,2,4}(n) =  13,  19,   61,   67,   73,   79, ... ~  (9*n/2)*log(n)
       {1,3,5}(n) =  31, 223,  229,  277,  283,  397, ... ~   (27*n)*log(n)
       {1,3,7}(n) =  43, 433,  457,  691, 1069, 1471, ... ~ (81*n/2)*log(n)
       {1,3,9}(n) = 109, 127,  157,  601,  733,  739, ... ~ (81*n/4)*log(n)
       {1,5,7}(n) = 307, 919, 1093, 2179, 2251, 3181, ... ~   (81*n)*log(n)
Note that the k-th q value takes A054272(k) x values and that a(n) = A040034(n) \ {1,2,4}(n). Following a result of Erdős (cf. A053760, A098990) the asymptotic means for q and x are Sum_{n>=1} prime(n)*2/3^n = 2.69463670741804726229622... and Sum_{n>=1} Sum_{prime(n) < k prime < prime(n)^2 OR k = prime(n)^2} D(prime(n),k)*k = 5.69767191389790422108748...
Subsequence of A040034 (2 is not a cubic residue modulo p) such that 3 is neither a residue nor in the same cubic power class as 2. (End)

			From _Travis Scott_, Oct 04 2022: (Start)
{1,2,3}^12 (mod 37) == {1,26,10} covers the 12th-power residues on Z/37Z.
{1,2,3}^14 (mod 43) == {1,1,36} misses 6. (End)

S. Saidi, Semicrosses and quadratic forms, European J. Combinatorics, vol.16, pp. 191-196, 1995. [This article has a typesetting error at the last term of this sequence on Table 1, p. 195. - _Travis Scott_, Oct 04 2022]

Subsequence of A040034.

Select[Prime@Range@497,Mod[#,3]==1&&DuplicateFreeQ@PowerMod[{1,2,3},(#-1)/3,#]&] (* Travis Scott, Oct 04 2022 *)

From Travis Scott, Oct 04 2022: (Start)
Primes of quadratic form 7x^2 +- 6xy + 36y^2 [from Saidi].
a(n) ~ 9*n*log(n). (End)

Incorrect term deleted and more terms from Travis Scott, Oct 04 2022

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

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A259189 Semiprimes of the form n^3 + 2.

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A107662 -n is the discriminant of cubic polynomials irreducible over Zp for primes p represented by only one binary quadratic form.

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A162541 Primes p such that a splitting of the cyclic group Zp by the perfect 3-shift code {+-1,+-2,+-3} exists.

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