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A162541
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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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0
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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
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OFFSET
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1,1
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COMMENTS
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This list was computed by S. Saidi.
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)
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LINKS
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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]
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FORMULA
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Primes of quadratic form 7x^2 +- 6xy + 36y^2 [from Saidi].
a(n) ~ 9*n*log(n). (End)
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EXAMPLE
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{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)
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MATHEMATICA
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Select[Prime@Range@497, Mod[#, 3]==1&&DuplicateFreeQ@PowerMod[{1, 2, 3}, (#-1)/3, #]&] (* Travis Scott, Oct 04 2022 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Incorrect term deleted and more terms from Travis Scott, Oct 04 2022
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STATUS
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approved
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