A346277 Primitive terms of A108569.

4, 110, 506, 550, 1830, 2162, 2750, 3422, 4114, 4746, 5490, 5566, 6806, 7782, 9150, 11342, 13750, 14238, 16470, 16762, 23346, 27450, 27722, 31862, 33222, 42714, 43378, 45254, 45750, 49410, 49726, 51302, 61226, 68750, 68906, 70038, 82350, 99238, 99666, 112110, 115650

Offset: 1

Author

Bernard Schott, Aug 22 2021

Keywords

• nonn

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, ...

Examples

			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.
		

Crossrefs

Cf. A000010, A005384.

Subsequence of A108569.

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

Programs

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:

PARI

f(m) = eulerphi(m+eulerphi(m)) - eulerphi(m);
isok(m) = !f(m) && !(m % 2) && f(m/2); \\ Michel Marcus, Aug 31 2021