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A049115
a(n) is the number of iterations of the Euler phi function needed to reach a power of 2, when starting from n.
7
0, 0, 1, 0, 1, 1, 2, 0, 2, 1, 2, 1, 2, 2, 1, 0, 1, 2, 3, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 1, 2, 0, 2, 1, 2, 2, 3, 3, 2, 1, 2, 2, 3, 2, 2, 3, 4, 1, 3, 2, 1, 2, 3, 3, 2, 2, 3, 3, 4, 1, 2, 2, 3, 0, 2, 2, 3, 1, 3, 2, 3, 2, 3, 3, 2, 3, 2, 2, 3, 1, 4, 2, 3, 2, 1, 3, 3, 2, 3, 2, 3, 3, 2, 4, 3, 1, 2, 3, 2, 2, 3, 1, 2, 2, 2
OFFSET
1,7
COMMENTS
a(n) = A227944(n) if n is not a power of 2. - Eric M. Schmidt, Oct 13 2013
FORMULA
The smallest x so that Nest[ EulerPhi, n, x ] = 2^w is just achieved.
From Antti Karttunen, Aug 28 2021: (Start)
If A209229(n) = 1, then a(n) = 0, otherwise a(n) = 1 + a(A000010(n)).
a(n) <= A003434(n) and a(n) <= A329697(n) for all n.
(End)
EXAMPLE
If n is a power of 2, then a(n)=0 by definition. If n = 59049, then by iterating with phi, we get 59049 -> 39366 -> 13122 -> 4374 -> 1458 -> 486 -> 162 -> 54 -> 18 -> 6 -> 2 -> 1. It took ten steps to reach the first power of 2 (2 in this case), so a(59049) = 10.
MATHEMATICA
Table[If[IntegerQ@ Log2@ n, 0, -1 + Length@ NestWhileList[EulerPhi, n, ! IntegerQ@ Log2@ # &]], {n, 105}] (* Michael De Vlieger, Aug 01 2017 *)
PROG
(PARI) A049115(n) = if(!bitand(n, n-1), 0, 1+A049115(eulerphi(n))); \\ Antti Karttunen, Aug 28 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Definition corrected and simplified, example corrected by Antti Karttunen, Aug 28 2021
STATUS
approved