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

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

Offset: 1

Author

David L. Harden, Mar 24 2018

Keywords

• nonn
• more

Comments

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.

No odd terms are in the sequence except for 1.

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

From Jianing Song, Dec 12 2021: (Start)

Conjecture: defining this sequence as "positive integers m such that whenever n > 1 is in the range of the Euler totient function, so is m*n" would give the same terms. That is to say, it seems that if m is a nontotient number, then there exists a totient number n > 1 such that m*n is a nontotient.

The known primitive terms of this sequence (terms that are not products of two previous terms) are 1, 2, 6, 18, 20. More terms are needed to determine the primitive terms further. (End)

Examples

			1 is trivially in the sequence.
Note that any value assumed by phi is assumed at an even argument, since k odd implies phi(k) = phi(2k).
Then 2 is in the sequence, since n = phi(k) and k even imply that 2n = phi(2k).
3 is not in the sequence: 30 = phi(31), but 3*30 = 90 is not in the range of phi.
4 is in the sequence because 2 is (using closure under multiplication).
5 is not in the sequence: 18 = phi(19), but 5*18 = 90 is not in the range of phi.
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.
7 is not in the sequence: 22 = phi(23), but 7*22 = 154 is not in the range of phi.
8 is in the sequence because 2 is.
		

Links

• Math Overflow, The range of the Euler totient function and multiplication by 28, 2018.

Crossrefs

Cf. A000010, A002202, A007617.