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A301587 Positive integers m such that whenever n is in the range of the Euler totient function, so is m*n. 1

%I

%S 1,2,4,6,8,12,16,18,20,24

%N Positive integers m such that whenever n is in the range of the Euler totient function, so is m*n.

%C Closure under multiplication: if multiplication by m_1 carries totient values to totient values and multiplication by m_2 does also, then so does their composition, which is multiplication by m_1*m_2.

%C No odd terms are in the sequence except for 1.

%C 32, 36, 40, 42, 48, 54, 64, and 72 are also in this sequence, although determining their position is difficult. - _Charlie Neder_, Aug 04 2019

%e 1 is trivially in the sequence.

%e Note that any value assumed by phi is assumed at an even argument, since k odd implies phi(k) = phi(2k).

%e Then 2 is in the sequence, since n = phi(k) and k even imply that 2n = phi(2k).

%e 3 is not in the sequence: 30 = phi(31), but 90 is not in the range of phi.

%e 4 is in the sequence because 2 is (using closure under multiplication).

%e 5 is not in the sequence: 18 = phi(19), but 90 is not in the range of phi.

%e 6 is in the sequence: If n = phi(k) with k even, phi(9k) = 6n if k is a nonmultiple of 3. If k is a multiple of 3, then 6n = phi(6k) since k is a multiple of 6.

%e 7 is not in the sequence: 22 = phi(23), but 154 is not in the range of phi.

%e 8 is in the sequence because 2 is.

%K nonn,more

%O 1,2

%A _David L. Harden_, Mar 24 2018

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Last modified May 31 02:51 EDT 2020. Contains 334747 sequences. (Running on oeis4.)