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%I #57 Jul 13 2021 19:49:37
%S 1,2,6,4,3,10,12,8,5,14,18,9,7,15,20,24,16,30,22,11,21,26,28,36,42,13,
%T 34,40,48,32,60,17,38,54,27,19,33,44,50,25,66,46,23,35,39,52,45,56,72,
%U 90,84,78,70,58,29,62,31,51,68,80,96,64,120
%N Numbers j such that the numbers Phi(j, m) are in sorted order for any integer m >= 2, where Phi(k, x) is the k-th cyclotomic polynomial.
%C Based on A002202 "Values taken by totient function phi(m)", A000010 can only take certain even numbers. So for the worst case, the largest Phi(k,m) with degree d (even positive integer) will be (1-k^(d+1))/(1-k) (or smaller) and the smallest Phi(k,m) with degree d+2 will be (1+k^(d+3))/(1+k) (or larger).
%C Note that (1+k^(d+3))/(1+k)-(1-k^(d+1))/(1-k) = (k/(k^2-1))*(2+k^d*(k^3-(k^2+k+1))) >= 0 since k^3 > k^2+k+1 when k >= 2.
%C This means that this sequence can be segmented into sets in which Phi(k,m) shares the same degree of polynomial and it can be generated in this way.
%H S. P. Glasby, <a href="https://arxiv.org/abs/1903.02951">Cyclotomic ordering conjecture</a>, arXiv:1903.02951 [math.NT], 2019.
%H Carl Pomerance and Simon Rubinstein-Salzedo, <a href="https://arxiv.org/abs/1903.01962">Cyclotomic Coincidences</a>, arXiv:1903.01962 [math.NT], 2019.
%e For k such that A000010(k) = 1,
%e Phi(1,m) = -1 + m,
%e Phi(2,m) = 1 + m,
%e Phi(1,m) < Phi(2,m),
%e so, a(1)=1, a(2)=2.
%e For k > 2 such that A000010(k) = 2,
%e Phi(3,m) = 1 + m + m^2,
%e Phi(4,m) = 1 + m^2,
%e Phi(6,m) = 1 - m + m^2.
%e For m > 1, Phi(6,m) < Phi(4,m) < Phi(3,m), so a(3)=6, a(4)=4, and a(5)=3 (noting that Phi(6,m) > Phi(2,m) when m > 2, and Phi(6,2) = Phi(2,2)).
%e For k such that A000010(k) = 4,
%e Phi(5,m) = 1 + m + m^2 + m^3 + m^4,
%e Phi(8,m) = 1 + m^4,
%e Phi(10,m) = 1 - m + m^2 - m^3 + m^4,
%e Phi(12,m) = 1 - m^2 + m^4.
%e For m > 1, Phi(10,m) < Phi(12,m) < Phi(8,m) < Phi(5,m), so a(6) = 10, a(7) = 12, a(8) = 8, and a(9) = 5 (noting Phi(10,m) - Phi(3,m) = m((m^2 + m + 2)(m - 2) + 2) >= 4 > 0 when m >= 2).
%t t = Select[Range[400], EulerPhi[#] <= 40 &]; SortBy[t, Cyclotomic[#, 2] &]
%Y Cf. A194712, A000010, A002202, A032447.
%K nonn
%O 1,2
%A _Lei Zhou_, Feb 13 2012
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