This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A301587 #32 Apr 14 2023 06:47:09 %S A301587 1,2,4,6,8,12,16,18,20,24 %N A301587 Positive integers m such that whenever n is in the range of the Euler totient function, so is m*n. %C A301587 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 A301587 No odd terms are in the sequence except for 1. %C A301587 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 %C A301587 From _Jianing Song_, Dec 12 2021: (Start) %C A301587 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. %C A301587 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) %H A301587 Math Overflow, <a href="https://mathoverflow.net/questions/307392/the-range-of-the-euler-totient-function-and-multiplication-by-28">The range of the Euler totient function and multiplication by 28</a>, 2018. %e A301587 1 is trivially in the sequence. %e A301587 Note that any value assumed by phi is assumed at an even argument, since k odd implies phi(k) = phi(2k). %e A301587 Then 2 is in the sequence, since n = phi(k) and k even imply that 2n = phi(2k). %e A301587 3 is not in the sequence: 30 = phi(31), but 3*30 = 90 is not in the range of phi. %e A301587 4 is in the sequence because 2 is (using closure under multiplication). %e A301587 5 is not in the sequence: 18 = phi(19), but 5*18 = 90 is not in the range of phi. %e A301587 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 A301587 7 is not in the sequence: 22 = phi(23), but 7*22 = 154 is not in the range of phi. %e A301587 8 is in the sequence because 2 is. %Y A301587 Cf. A000010, A002202, A007617. %K A301587 nonn,more %O A301587 1,2 %A A301587 _David L. Harden_, Mar 24 2018