A176841 a(n) is the number of iterations of f(n) = n-phi(tau(n)) needed to reach 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

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

Cf. A000005, A000010.

Sequence in context: A238263 A071542 A264810 * A176814 A088461 A135020

Adjacent sequences: A176838 A176839 A176840 * A176842 A176843 A176844

Keyword

nonn

Author

Michel Lagneau, Apr 27 2010

Status

approved