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Primitive terms of A108569.
0

%I #41 Dec 23 2024 22:48:56

%S 4,110,506,550,1830,2162,2750,3422,4114,4746,5490,5566,6806,7782,9150,

%T 11342,13750,14238,16470,16762,23346,27450,27722,31862,33222,42714,

%U 43378,45254,45750,49410,49726,51302,61226,68750,68906,70038,82350,99238,99666,112110,115650

%N Primitive terms of A108569.

%C If k is an even term of A108569 then 2k is another term.

%C This sequence lists the initial term k_0 of each infinite subsequence of A108569 that is solution of the equation phi(k) = phi(k + phi(k)).

%C As 2 is no solution, A108569(1) = 1 is not primitive.

%C Each k_0 > 4 is of the form k_0 = 2*m with m odd.

%C If p > 3 is a Sophie Germain prime, then every m = 2*p^q*(2p+1), q >=1 is a term because in this case, phi(m) = phi(m+phi(m)) = 2*(p-1)*p^q; the first terms that are not of this form are 4, 1830, 4114, ...

%e a(1) = 4 because every k = 2^m, m >= 2 satisfies phi(k) = phi(k+phi(k)) = 2^(m-1), and k_0 = 4 is the smallest term of this subsequence of A108569.

%e a(2) = 110 because every k = 5*11*2^m, m >= 1 satisfies phi(k) = phi(k+phi(k)) = 5*2^(m+2) and k_0 = 110 is the smallest term of this subsequence of A108569 (note that 5 is a Sophie Germain prime).

%e a(5) = 1830 because every k = 3*5*61*2^m, m >= 1 satisfies phi(k) = phi(k+phi(k)) = 3*5*2^(m+4) and k_0 = 1830 is the smallest term of this subsequence of A108957.

%p with(numtheory):

%p for m from 2 to 116000 by 2 do

%p u:=phi(m+phi(m)) - phi(m);

%p if u=0 and phi(m/2 + phi(m/2)) <> phi(m/2) then print(m); else fi; od:

%o (PARI) f(m) = eulerphi(m+eulerphi(m)) - eulerphi(m);

%o isok(m) = !f(m) && !(m % 2) && f(m/2); \\ _Michel Marcus_, Aug 31 2021

%Y Cf. A000010, A005384.

%Y Subsequence of A108569.

%Y Similar to A346694 (with phi(k) = phi(k-phi(k))).

%K nonn

%O 1,1

%A _Bernard Schott_, Aug 22 2021