A023137 Number of cycles of function f(x) = 5x mod n.

1, 2, 2, 4, 1, 4, 2, 6, 3, 2, 3, 8, 4, 4, 2, 8, 2, 6, 3, 4, 5, 6, 2, 14, 1, 8, 4, 8, 3, 4, 11, 10, 6, 4, 2, 12, 2, 6, 11, 6, 3, 10, 2, 12, 3, 4, 2, 20, 3, 2, 5, 16, 2, 8, 3, 14, 6, 6, 3, 8, 3, 22, 12, 12, 4, 12, 4, 8, 5, 4, 15, 22, 2, 4, 2, 12, 6, 22, 3, 8, 5, 6, 2, 20, 2, 4, 8, 18, 3, 6, 11, 8, 22, 4, 3, 26

Offset

1,2

Comments

Number of factors in the factorization of the polynomial x^n-1 over GF(5). - T. D. Noe, Apr 16 2003

References

R. Lidl and H. Niederreiter, Finite Fields, Addison-Wesley, 1983, p. 65.

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

Formula

a(n) = Sum_{d|m} phi(d)/ord(5, d), where m is n with all factors of 5 removed. - T. D. Noe, Apr 19 2003

a(n) = (1/ord(5,m))*Sum_{j = 0..ord(5,m)-1} gcd(5^j - 1, m), where m is n with all factors of 5 removed. - Nihar Prakash Gargava, Nov 14 2018

Example

a(15) = 2 because (1) the function 5x mod 15 has the two cycles (0),(5,10) and (2) the factorization of x^15-1 over integers mod 5 is (4+x)^5 (1+x+x^2)^5, which has two unique factors. Note that the length of the cycles is the same as the degree of the factors.

Mathematica

Table[Length[FactorList[x^n - 1, Modulus -> 5]] - 1, {n, 100}]
CountFactors[p_, n_] := Module[{sum=0, m=n, d, f, i}, While[Mod[m, p]==0, m/=p]; d=Divisors[m]; Do[f=d[[i]]; sum+=EulerPhi[f]/MultiplicativeOrder[p, f], {i, Length[d]}]; sum]; Table[CountFactors[5, n], {n, 100}]

Prog

(PARI) a(n)={sumdiv(n/5^valuation(n, 5), d, eulerphi(d)/znorder(Mod(5, d))); }
vector(100, n, a(n)) \\ Joerg Arndt, Jan 22 2024
(Python)
from sympy import totient, n_order, divisors
def A023137(n):
    a, b = divmod(n, 5)
    while not b:
        n = a
        a, b = divmod(n, 5)
    return sum(totient(d)//n_order(5, d) for d in divisors(n, generator=True) if d>1)+1 # Chai Wah Wu, Apr 09 2024

Crossrefs

Cf. A000005, A000374.

Cf. A023135, A023136, A023138, A023139, A023140, A023141, A023142.

Sequence in context: A248503 A359855 A214714 * A327488 A065273 A352707

Adjacent sequences: A023134 A023135 A023136 * A023138 A023139 A023140

Keyword

nonn,changed

Author

David W. Wilson

Status

approved