A053039 Exponent of largest power of 2 which appears in the cototient-iteration started with n!.

0, 1, 2, 4, 5, 7, 9, 12, 14, 11, 13, 16, 14, 16, 23, 27, 23, 23, 24, 21, 23, 28, 42, 46, 26, 26, 26, 36, 43, 29, 50, 55, 37, 37, 40, 40, 39, 59, 39, 44, 68, 42, 42, 44, 51, 45, 50, 53, 49, 52, 51, 85, 55, 57, 53, 57, 60, 85, 62, 71, 62, 63, 60, 66, 66, 107, 67, 101, 76, 70, 75, 77

Offset

1,3

Comments

If the exponent is a(n), then the number of powers of 2 in the iteration-chain is 1+a(n), the maximal 2-power is 2^a(n) and the number of iterations (until fixed state) performed on these 2-powers is a(n).

Links

Table of n, a(n) for n=1..72.

Example

For n = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and starting the iteration of A051593 with n!, the first powers of 2 which appear are 1, 2, 4, 16, 32, 128, 512, 4096, 16384, 2048 and the corresponding exponents are a(n) = 0, 1, 2, 4, 5, 7, 9, 12, 14, 11.

Mathematica

Log2 /@ Table[NestWhile[# - EulerPhi@ # &, n!, ! IntegerQ@ Log2@ # &], {n, 60}] (* Michael De Vlieger, Aug 15 2017 *)

Prog

(PARI) cototient(x)= x - eulerphi(x)
FunctionIterate(f, x, t)= {local(retval); retval = vector(0); while(x!=t, x = eval(concat(f, "(x)")); retval = concat(retval, x)); retval; }
A053039(x) = {local(li, fa, retval); count = 0; li = concat([x! ], FunctionIterate("cototient", x!, 0)); for(i=1, #li, fa = factor(li[i]); if(((matsize(fa)[1] == 1) && (fa[1, 1] == 2)), retval = fa[1, 2]; break)); retval}
for(i=1, 72, print1(A053039(i), ", ")) \\ Olaf Voß, Feb 21 2008

Crossrefs

Cf. A051953, A053038, A053475.

Sequence in context: A007818 A158618 A000788 * A286753 A325543 A214051

Adjacent sequences: A053036 A053037 A053038 * A053040 A053041 A053042

Keyword

nonn

Author

Labos Elemer, Feb 24 2000

Extensions

More terms from Olaf Voß, Feb 21 2008

Status

approved