|
|
A156776
|
|
Number of iterations of x->(sigma(x)+phi(x))/2 until a non-integer is reached when starting with x=n; a(n)=0 if this never happens.
|
|
2
|
|
|
0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 3, 2, 1, 0, 1, 0, 2, 0, 0, 0, 4, 1, 0, 0, 4, 0, 0, 0, 1, 4, 3, 2, 1, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 7, 1, 1, 0, 0, 0, 8, 3, 2, 0, 0, 0, 0, 0, 8, 7, 1, 0, 0, 0, 0, 7, 6, 0, 1, 0, 0, 0, 4, 6, 5, 0, 0, 1, 0, 0, 5, 6, 5, 4, 3, 0, 9, 0, 0, 7, 6, 5, 4, 0, 1, 9, 1, 0, 5, 0, 9, 3
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,12
|
|
COMMENTS
|
In [Guy 1997] the iteration is said to fracture when sigma(x)+phi(x) becomes odd. For n with a(n)=0, A156775(n) gives the number of iterations until a previously seen term is encountered.
|
|
LINKS
|
|
|
EXAMPLE
|
Let f(x)=(sigma(x)+phi(x))/2. For x=1 we have f(x) = (1+1)/2 = 1, i.e. this is a fixed point and the sequence will never fraction, hence a(1)=0. The same happens for x=2, x=3 and x=5. For x=4 we have f(x) = (7+2)/2 = 9/2, the sequence "fractures" after a(4)=1 iterations. For x=6 we have f(x) = (12+2)/2 = 7, f(7) = (8+6)/2 = 7, a fixed point, so again a(6)=a(7)=0.
|
|
PROG
|
(PARI) A156776(n, u=[])={ until( denominator( n=(sigma(n)+eulerphi(n))/2)>1 | setsearch(u, n), u=setunion(u, Set(n))); if( denominator(n)>1, #u) }
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|