OFFSET
1,3
LINKS
T. D. Noe, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = Sum_{d|m} phi(d)/ord(4, d), where m is n with all factors of 2 removed. The formula was developed by extending the ideas in A000374 to composite multipliers. - T. D. Noe, Apr 21 2003
Mobius transform of A133702: (1, 2, 4, 3, 4, 8, 4, 4, 9, 8, ...). = Row sums of triangle A133703. - Gary W. Adamson, Sep 21 2007
a(n) = (1/ord(4, m))*Sum_{j = 0..ord(4, m) - 1} gcd(4^j - 1, m), where m is the odd part of n (A000265). - Nihar Prakash Gargava, Nov 14 2018
EXAMPLE
a(9) = 5 because the function 4x mod 9 has the five cycles (0),(3),(6),(1,4,7),(2,8,5).
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[4, n], {n, 100}]
PROG
(PARI) a(n)=sumdiv(n>>valuation(n, 2), d, eulerphi(d)/znorder(Mod(4, d))) \\ Charles R Greathouse IV, Aug 05 2016
(Python)
from sympy import totient, n_order, divisors
def A023136(n): return sum(totient(d)//n_order(4, d) for d in divisors(n>>(~n & n-1).bit_length(), generator=True) if d>1)+1 # Chai Wah Wu, Apr 09 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved