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A307436 a(n) is the smallest k such that (Z/kZ)* contains C_(2n) X C_(2n) as a subgroup, where (Z/kZ)* is the multiplicative group of integers modulo n. 2

%I

%S 8,65,63,544,275,455,1247,1088,513,1025,1541,7081,4187,3277,1891,

%T 12416,14111,2701,43739,7667,2107,10235,6533,11543,12625,8321,8829,

%U 31753,13747,8723,116003,49408,10787,39593,14981,23579,33227,104653,12403,45067,61337

%N a(n) is the smallest k such that (Z/kZ)* contains C_(2n) X C_(2n) as a subgroup, where (Z/kZ)* is the multiplicative group of integers modulo n.

%C a(n) exists for all n: by Dirichlet's theorem on arithmetic progressions, there exists two primes p, q congruent to 1 modulo 2n, in which case C_(2n) X C_(2n) is a subgroup of (Z/(p*q)Z)*.

%C a(n) is the smallest k such that there exists some x, y such that ord(x,k) = ord(y,k) = 2n and the set of powers of x and the set of powers of y modulo k have trivial intersection {1}, where ord(x,k) is the multiplicative order of x modulo n.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Multiplicative_group_of_integers_modulo_n ">Multiplicative group of integers modulo n</a>

%e (Z/8Z)* = C_2 X C_2, in which {3^e mod 8} = {3, 1} and {3^e mod 8} = {5, 1};

%e (Z/65Z)* = C_4 X C_12, in which {8^e mod 65} = {8, 64, 57, 1} and {12^e mod 65} = {12, 14, 38, 1};

%e (Z/63Z)* = C_6 X C_6, in which {2^e mod 63} = {2. 4. 8, 16, 32, 1} and {5^e mod 63} = {5, 25, 62, 58, 38, 1};

%e (Z/544Z)* = C_2 X C_8 X C_16, in which {9^e mod 544} = {9, 81, 185, 33, 297, 497, 121, 1} and {13^e mod 544} = {13, 169, 21, 273, 285, 441, 293, 1};

%e (Z/275Z)* = C_10 X C_20, in which {4^e mod 275} = {4, 16, 64, 256, 199, 246, 159, 86, 69, 1} and {6^e mod 275} = {6, 36, 216, 196, 76, 181, 261, 191, 46, 1};

%e (Z/455Z)* = C_2 X C_12 X C_12, in which {2^e mod 455} = {2, 4, 8, 16, 32, 64, 128, 256, 57, 114, 228, 1} and {3^e mod 455} = {3, 9, 27, 81, 243, 274, 367, 191, 118, 354, 152, 1}.

%o (PARI) a(n) = my(d=1, v=znstar(d)[2]); while(sum(i=1, #v, !(v[i]%(2*n)))<2, d++; v=znstar(d)[2]); d

%Y Cf. A307437.

%K nonn

%O 1,1

%A _Jianing Song_, Apr 08 2019

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Last modified September 20 16:48 EDT 2019. Contains 327242 sequences. (Running on oeis4.)