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A176841
a(n) is the number of iterations of f(n) = n-phi(tau(n)) needed to reach 1.
1
0, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 7, 8, 8, 9, 9, 10, 10, 11, 10, 12, 11, 13, 12, 13, 12, 13, 13, 14, 14, 15, 13, 14, 14, 15, 14, 15, 15, 16, 16, 17, 17, 18, 17, 19, 18, 20, 19, 20, 19, 21, 20, 22, 21, 22, 21, 22, 22, 23, 22, 24, 23, 24, 24, 25, 24, 25, 25, 26, 26, 27, 27
OFFSET
1,3
COMMENTS
tau(n) is the number of divisors of n (A000005) and phi(n) is the Euler totient function (A000010).
REFERENCES
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
LINKS
EXAMPLE
for n = 13, the number 7 is in the sequence because :
f(13) = 13 - phi(tau(13)) = 13 - phi(2) = 13 - 1 = 12;
f(12) = 12 - phi(tau(12)) = 12 - phi(6) = 12 - 2 = 10;
f(10) = 10 - phi(tau(10)) = 10 - phi(4) = 10 - 2 = 8;
f(8) = 8- phi(tau(8)) = 8 - phi(4) = 8 - 2 = 6 ;
f(6) = 6- phi(tau(6)) = 6 - phi(4) = 6 - 2 = 4 ;
f(4) = 4- phi(tau(4)) = 4 - phi(3) = 4 - 2 = 2 ;
f(2) = 2- phi(tau(2)) = 2 - phi(2) = 2 - 1 = 1, and a(13) = 7.
MAPLE
with(numtheory): n0:=200:tabl:=array(1..n0): for n from 1 to 1000 do:k:=0:nn:=n:for q from 0 to 1000 while(nn<>1) do:nn:=nn - phi(tau((nn))):k:=k+1:od:tabl[n]:=k:od:print(tabl):
MATHEMATICA
f[n_] := (k++; n - EulerPhi[ DivisorSigma[0, n]]); f[0] = 0; a[n_] := (k=0; FixedPoint[f, n]; k-1); Table[a[n], {n, 1, 76}](* Jean-François Alcover, May 10 2012 *)
CROSSREFS
Sequence in context: A238263 A071542 A264810 * A176814 A088461 A135020
KEYWORD
nonn
AUTHOR
Michel Lagneau, Apr 27 2010
STATUS
approved