A097029 Fixed points when the function f(x) = phi(x) + floor(x/2) is iterated, i.e., solutions to f(x) = x.

1, 2, 3, 4, 8, 15, 16, 32, 64, 128, 255, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65535, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 83623935, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967295

Offset

1,2

Comments

Trivial fixed points are the powers of 2. How many nontrivial cases exist like 3, 15, 255, 65535: the first 5 terms of A051179. More?

83623935 is the next such term (see also A050474 and A203966). - Michel Marcus, Nov 13 2015

Links

Max Alekseyev, Table of n, a(n) for n = 1..91

Example

For fixed points the cycle lengths are A097026(n=fix)=1, but the reverse is not true because long transients may also lead to 1-cycles.
So, e.g., 1910 is not here because its terminal 1-cycle is prefixed by a long transient: {1910, 1715, 2033, 2924, 2806, 2723, 3689, 4724, 4722, 3933, 4342, 4163, 6041, 8192, 8192}.

Prog

(PARI) isok(n) = eulerphi(n) + n\2 == n; \\ Michel Marcus, Nov 13 2015

Crossrefs

Union of A000079 and A050474.

Cf. A000010, A051179, A097026, A097027, A097028.

Sequence in context: A347754 A005853 A161460 * A122774 A274166 A364595

Adjacent sequences: A097026 A097027 A097028 * A097030 A097031 A097032

Keyword

nonn

Author

Labos Elemer, Aug 27 2004

Extensions

a(30)-a(35) from Michel Marcus, Nov 13 2015

a(36)-a(38) from Jinyuan Wang, Jul 22 2021

Status

approved