A037226 a(n) = phi(2n+1) / multiplicative order of 2 mod 2n+1.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

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^n-1. 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 Rudin-Shapiro polynomials, J. Reine Angew. Math.288 (1976), 37--65. See Table 2. MR0498479 (58 #16589).
• Jarkko Peltomäki and Aleksi Saarela, Standard words and solutions of the word equation X_1^2 ... X_n^2 = (X_1 ... X_n)^2, Journal of Combinatorial Theory, Series A (2021) Vol. 178, 105340. See also arXiv:2004.14657 [cs.FL], 2020.

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).

Programs

Mathematica

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

PARI

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

Formula

a(n) = Sum{d|2n+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).