OFFSET
1,1
COMMENTS
If k is an even term of A108569 then 2k is another term.
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)).
As 2 is no solution, A108569(1) = 1 is not primitive.
Each k_0 > 4 is of the form k_0 = 2*m with m odd.
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, ...
EXAMPLE
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.
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).
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.
MAPLE
with(numtheory):
for m from 2 to 116000 by 2 do
u:=phi(m+phi(m)) - phi(m);
if u=0 and phi(m/2 + phi(m/2)) <> phi(m/2) then print(m); else fi; od:
PROG
(PARI) f(m) = eulerphi(m+eulerphi(m)) - eulerphi(m);
isok(m) = !f(m) && !(m % 2) && f(m/2); \\ Michel Marcus, Aug 31 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Bernard Schott, Aug 22 2021
STATUS
approved