

A206710


This irregular table contains indices j, k, l,... in each row such that the values Phi(j,m) < Phi(k,m)< Phi(l,m)< ... of cyclotomic polynomials Phi(.,.) are sorted given any constant integer argument m >= 2.


1



1, 2, 3, 4, 6, 5, 12, 8, 10, 7, 9, 18, 14, 30, 20, 24, 16, 15, 11, 22, 42, 13, 28, 36, 21, 26, 17, 40, 48, 32, 60, 34, 19, 27, 54, 38, 66, 44, 25, 50, 33, 23, 46, 70, 78, 52, 90, 56, 72, 45, 84, 39, 35, 29, 58, 31, 62, 102, 68, 80, 96, 64, 120
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OFFSET

1,2


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 (1k^(d+1))/(1k) (or smaller)and the smallest Phi(k,m) with degree d+2 will be (1+k^(d+3))/(1+k) (or larger).
(1+k^(d+3))/(1+k)(1k^(d+1))/(1k)=(k/(k^21))*(2+k^d*(k^3(k^2+k+1)))
k^3>k^2+k+1 when k>=2.
This means that this sequence can be segmented to sets in which Cyclotomic(k,m) shares the same degree of Polynomial and it can be generated in this way.


LINKS

Table of n, a(n) for n=1..63.


EXAMPLE

For those k's that make A000010(k) = 1
Phi(1,m) = 1m
Phi(2,m) = 1m
Phi(1,m) < Phi(2,m)
So, a(1) = 1, a(2) = 2;
For those k's (k > 2) that make A000010(k) = 2
Phi(3,m) = 1  m + m^2
Phi(4,m) = 1 + m^2
Phi(6,m) = 1 + m + m^2
Obviously when integer m > 1, Phi(3,m) < Phi(4,m) < Phi(6,m)
So a(3)=3, a(4)=4, and a(5)=6
For those k's that make 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
Obviously when integer m > 1, Phi(5,m) < Phi(12,m) < Phi(8,m) < Phi(10,m),
So a(6) = 5, a(7) = 12, a(8) = 8, and a(9) = 10.
The table starts
1,2;
3,4,6;
5,12,8,10;


MATHEMATICA

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


CROSSREFS

Cf. A206292, A194712, A206225, A000010, A032447.
Sequence in context: A064787 A075160 A119755 * A175502 A240809 A214322
Adjacent sequences: A206707 A206708 A206709 * A206711 A206712 A206713


KEYWORD

nonn,tabf


AUTHOR

Lei Zhou, Feb 13 2012


STATUS

approved



