A176844 - OEIS

A176844

The number of iterations of the map n -> n - bigomega(sigma(n)) until reaching 1.

0, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 11, 10, 11, 11, 12, 12, 12, 12, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 16, 15, 16, 16, 16, 17, 16, 17, 17, 18, 18, 19, 17, 19, 20, 20, 19, 21, 21, 20, 21, 20, 21, 21, 21, 22, 22, 21, 22, 22

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OFFSET

1,4

COMMENTS

The function n-bigomega(sigma(n)) = n - A001222(A000203(n)) = 1, 1, 1, 3, 3, 3, 4, 6, 8, 7, 8, 9, 11, 10, 11, 15, 14, 16, 16... (n>=1) is iterated until reaching one of the 1's.

LINKS

Table of n, a(n) for n=1..79.

EXAMPLE

a(n=12) = 5 because :

12 - bigomega(sigma(12)) = 9 (1st step); 9 - bigomega(sigma(9)) = 8 (2nd step)

8 - bigomega(sigma(8)) = 6 (3rd step); 6 - bigomega(sigma(6)) = 3 (4th step);

3 - bigomega(sigma(3)) = 1 (fifth and final step).

MAPLE

with(numtheory): n0:=200:tabl:=array(1..n0):

for n from 1 to n0 do: k:=0: nn:=n: for q from 0 to 1000 while(nn<>1) do: nn:=nn - bigomega(sigma((nn))): k:=k+1: od: tabl[n]:=k: od: print(tabl):

MATHEMATICA

Table[Length[NestWhileList[#-PrimeOmega[DivisorSigma[1, #]]&, n, #>1&]]- 1, {n, 80}] (* Harvey P. Dale, Dec 16 2012 *)