A162541 - OEIS

%I #30 Nov 22 2022 22:27:05

%S 7,37,139,163,181,241,313,337,349,379,409,421,541,571,607,631,751,859,

%T 877,937,1033,1087,1123,1171,1291,1297,1447,1453,1483,1693,1741,1747,

%U 2011,2161,2239,2311,2371,2473,2539,2647,2677,2707,2719,2857,3169,3361,3433,3511,3547

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

%C This list was computed by S. Saidi.

%C From _Travis Scott_, Oct 04 2022: (Start)

%C 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.

%C a(n)-> {1,2,3}(n) =   7,  37,  139,  163,  181,  241, ... ~    (9*n)*log(n)

%C        {1,2,4}(n) =  13,  19,   61,   67,   73,   79, ... ~  (9*n/2)*log(n)

%C        {1,3,5}(n) =  31, 223,  229,  277,  283,  397, ... ~   (27*n)*log(n)

%C        {1,3,7}(n) =  43, 433,  457,  691, 1069, 1471, ... ~ (81*n/2)*log(n)

%C        {1,3,9}(n) = 109, 127,  157,  601,  733,  739, ... ~ (81*n/4)*log(n)

%C        {1,5,7}(n) = 307, 919, 1093, 2179, 2251, 3181, ... ~   (81*n)*log(n)

%C 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...

%C 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)

%H S. Saidi, <a href="https://doi.org/10.1016/0195-6698(95)90058-6">Semicrosses and quadratic forms</a>, 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]

%F From _Travis Scott_, Oct 04 2022: (Start)

%F Primes of quadratic form 7x^2 +- 6xy + 36y^2 [from Saidi].

%F a(n) ~ 9*n*log(n). (End)

%e From _Travis Scott_, Oct 04 2022: (Start)

%e {1,2,3}^12 (mod 37) == {1,26,10} covers the 12th-power residues on Z/37Z.

%e {1,2,3}^14 (mod 43) == {1,1,36} misses 6. (End)

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

%Y Subsequence of A040034.

%K nonn

%O 1,1

%A _Ctibor O. Zizka_, Jul 05 2009

%E Incorrect term deleted and more terms from _Travis Scott_, Oct 04 2022