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A307437 a(n) is the smallest k such that 2n divides psi(k), psi = A002322. 2

%I

%S 3,5,7,17,11,13,29,17,19,25,23,73,53,29,31,97,103,37,191,41,43,89,47,

%T 97,101,53,81,113,59,61,311,193,67,137,71,73,149,229,79,187,83,203,

%U 173,89,181,235,283,97,197,101,103,313,107,109,121,113,229,233,709,241

%N a(n) is the smallest k such that 2n divides psi(k), psi = A002322.

%C a(n) exists for all n: by Dirichlet's theorem on arithmetic progressions, there exists a prime p congruent to 1 modulo 2n, so 2n divides psi(p) = p - 1.

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

%C a(n) is the smallest k such that there exists some x such that ord(x,k) = 2n, where ord(x,k) is the multiplicative order of x modulo k.

%C Record values of a(n)/n occur at n = 1, 4, 12, 19, 59, 167, 196, 197, 227, 317, 457, 521, 706, ...

%H Robert Israel, <a href="/A307437/b307437.txt">Table of n, a(n) for n = 1..10000</a>

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

%e For n = 7, psi(29) = 28 and 29 is the smallest k such that 14 divides psi(k), so a(7) = 29.

%e For n = 27, psi(81) = 54 and 81 is the smallest k such that 54 divides psi(k), so a(27) = 81.

%e For n = 40, psi(187) = 80 and 187 is the smallest k such that 80 divides psi(k), so a(40) = 187.

%e For n = 42, psi(203) = 84 and 203 is the smallest k such that 84 divides psi(k), so a(42) = 203.

%p N:= 100: # for a(1)..a(N)

%p V:= Vector(N): count:= 0:

%p for k from 3 while count < N do

%p S:= select(t -> t <= N and V[t]=0, numtheory:-divisors(numtheory:-lambda(k)/2));

%p if nops(S) > 0 then count:= count + nops(S); V[convert(S,list)]:= k fi

%p od:

%p convert(V,list); # _Robert Israel_, Jul 10 2019

%o (PARI) a(n) = my(i=1); while(A002322(i)%(2*n), i++); i \\ See A002322 for its program

%Y Cf. A002322, A307436.

%K nonn,look

%O 1,1

%A _Jianing Song_, Apr 08 2019

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Last modified October 15 20:04 EDT 2019. Contains 328037 sequences. (Running on oeis4.)