A206225 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.

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, 34, 40, 48, 32, 60, 17, 38, 54, 27, 19, 33, 44, 50, 25, 66, 46, 23, 35, 39, 52, 45, 56, 72, 90, 84, 78, 70, 58, 29, 62, 31, 51, 68, 80, 96, 64, 120

Offset: 1

Author

Lei Zhou, Feb 13 2012

Keywords

• nonn

Comments

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).

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.

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.

Examples

			For k such that A000010(k) = 1,
  Phi(1,m) = -1 + m,
  Phi(2,m) = 1 + m,
  Phi(1,m) < Phi(2,m),
  so, a(1)=1, a(2)=2.
For k > 2 such that A000010(k) = 2,
  Phi(3,m) = 1 + m + m^2,
  Phi(4,m) = 1 + m^2,
  Phi(6,m) = 1 - m + m^2.
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)).
For k such that A000010(k) = 4,
  Phi(5,m) = 1 + m + m^2 + m^3 + m^4,
  Phi(8,m) = 1 + m^4,
  Phi(10,m) = 1 - m + m^2 - m^3 + m^4,
  Phi(12,m) = 1 - m^2 + m^4.
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).
		

Links

• S. P. Glasby, Cyclotomic ordering conjecture, arXiv:1903.02951 [math.NT], 2019.
• Carl Pomerance and Simon Rubinstein-Salzedo, Cyclotomic Coincidences, arXiv:1903.01962 [math.NT], 2019.

Crossrefs

Cf. A194712, A000010, A002202, A032447.

Programs

Mathematica

t = Select[Range[400], EulerPhi[#] <= 40 &]; SortBy[t, Cyclotomic[#, 2] &]