

A037226


a(n) = phi(2n+1) / multiplicative order of 2 mod 2n+1.


4



1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 6, 2, 2, 1, 2, 2, 3, 2, 2, 2, 4, 1, 2, 2, 1, 1, 6, 4, 1, 2, 2, 8, 2, 2, 2, 1, 1, 8, 2, 8, 6, 6, 2, 2, 2, 1, 2, 4, 1, 3, 2, 4, 2, 6, 4, 1, 4, 1, 18, 6, 1, 6, 2, 2, 1, 2, 2, 4, 2, 1, 10, 4, 6, 3, 2, 4
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OFFSET

0,4


COMMENTS

Number of primitive irreducible factors of x^(2n+1)  1 over integers mod 2. There are no primitive irreducible factors for x^(2n)1 because it always has the same factors as x^n1. Considering that A000374 also counts the cycles in the map f(x) = 2x mod n, a(n) is also the number of primitive cycles of that mapping.  T. D. Noe, Aug 01 2003
Equals number of irreducible factors of the cyclotomic polynomial Phi(2n+1,x) over Z/2Z. All factors have the same degree.  T. D. Noe, Mar 01 2008


LINKS

T. D. Noe, Table of n, a(n) for n = 0..10000
Brillhart, John; Lomont, J. S.; Morton, Patrick. Cyclotomic properties of the RudinShapiro polynomials, J. Reine Angew. Math.288 (1976), 3765. See Table 2. MR0498479 (58 #16589).


FORMULA

a(n) = Sum{d2n+1} mu((2n+1)/d) A000374(d), the inverse Mobius transform of A000374  T. D. Noe, Aug 01 2003
a(n) = A037225(n)/A002326(n).


MATHEMATICA

a[n_] := EulerPhi[2n+1]/MultiplicativeOrder[2, 2n+1]; Table[a[n], {n, 0, 100}] (* JeanFrançois Alcover, Dec 10 2015 *)


PROG

(PARI) a(n)=eulerphi(2*n+1)/znorder(Mod(2, 2*n+1)) \\ Charles R Greathouse IV, Dec 29 2013


CROSSREFS

Cf. A000374 (number of irreducible factors of x^n  1 over integers mod 2), A081844.
Cf. A006694 (cyclotomic cosets of 2 mod 2n+1).
Sequence in context: A187759 A270645 A268059 * A089641 A086995 A220492
Adjacent sequences: A037223 A037224 A037225 * A037227 A037228 A037229


KEYWORD

nonn


AUTHOR

N. J. A. Sloane.


STATUS

approved



