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A023141 Number of cycles of function f(x) = 9x mod n. 7
1, 2, 1, 4, 3, 2, 3, 8, 1, 6, 3, 4, 5, 6, 3, 12, 3, 2, 3, 12, 3, 6, 3, 8, 5, 10, 1, 12, 3, 6, 3, 16, 3, 6, 9, 4, 5, 6, 5, 24, 11, 6, 3, 12, 3, 6, 3, 12, 5, 10, 3, 20, 3, 2, 9, 24, 3, 6, 3, 12, 13, 6, 3, 20, 15, 6, 7, 12, 3, 18, 3, 8, 13, 10, 5, 12, 9, 10, 3, 44, 1, 22, 3, 12, 13, 6, 3, 24, 3, 6, 31, 12, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

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

FORMULA

a(n) = Sum_{d|m} phi(d)/ord(9, d), where m is n with all factors of 3 removed. - T. D. Noe, Apr 21 2003

a(n) = (1/ord(9,m))*Sum_{j = 0..ord(9,m)-1} gcd(9^j - 1, m), where m is n with all factors of 3 removed. - Nihar Prakash Gargava, Nov 14 2018

EXAMPLE

a(12) = 4 because the function 9x mod 12 has the four cycles (0),(3),(1,9),(2,6).

MATHEMATICA

CountFactors[p_, n_] := Module[{sum=0, m=n, d, f, i, ps, j}, ps=Transpose[FactorInteger[p]][[1]]; Do[While[Mod[m, ps[[j]]]==0, m/=ps[[j]]], {j, Length[ps]}]; d=Divisors[m]; Do[f=d[[i]]; sum+=EulerPhi[f]/MultiplicativeOrder[p, f], {i, Length[d]}]; sum]; Table[CountFactors[9, n], {n, 100}]

CROSSREFS

Cf. A000374.

Cf. A023135, A023136, A023137, A023138, A023139, A023140, A023142.

Sequence in context: A183201 A082467 A106407 * A283324 A072650 A082497

Adjacent sequences:  A023138 A023139 A023140 * A023142 A023143 A023144

KEYWORD

nonn

AUTHOR

David W. Wilson

STATUS

approved

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Last modified September 26 08:09 EDT 2020. Contains 337346 sequences. (Running on oeis4.)