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

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

1,1

Comments

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 {1<q<x} exist in p == 1 (mod 3) for all q prime < x (prime or q^2) =< q^2 at a density of D(q,x) = 2^(pi(x) - pi(q) + [x = q^2]) / 3^(pi(x)), where pi(n) counts primes and [P] returns 1 if P else 0. Initial terms of the first few {1,q,x} are listed below.

a(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)

Links

Table of n, a(n) for n=1..49.

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]

Formula

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)

Example

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)

Mathematica

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

Crossrefs

Subsequence of A040034.

Sequence in context: A143991 A243893 A012885 * A196574 A094729 A202659

Adjacent sequences: A162538 A162539 A162540 * A162542 A162543 A162544

Keyword

nonn

Author

Ctibor O. Zizka, Jul 05 2009

Extensions

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

Status

approved