A053048 -id:A053048 - cp's OEIS frontend

A053047 a(n) is the first (and maximal) power of 2 arising during iterations of the Euler phi function with initial value n!.

1, 2, 2, 8, 32, 64, 128, 1024, 1024, 8192, 65536, 262144, 1048576, 4194304, 16777216, 268435456, 4294967296, 8589934592, 17179869184, 274877906944, 549755813888, 8796093022208, 140737488355328, 1125899906842624

Offset: 1

Labos Elemer, Feb 25 2000

nonn

			For n = 10, the initial value is 10! = 3628800 and the iteration chain is {3628800, 829440, 221184, 73728, 24576, 8192, 4096, 2048, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1}. The first power of 2 is the 6th element, arising after 5 iterations of phi, and its value is 8192.

Amiram Eldar, Table of n, a(n) for n = 1..657

Cf. A000010, A000142, A048855, A053038, A053048.

a(n) = 2^A053048(n). - Amiram Eldar, Aug 17 2024

A053099 When the Euler phi function is iterated with initial value A002110(n) = n-th primorial, a(n) = exponent of largest power of 2 arising in the iteration.

Original entry on oeis.org

1, 1, 3, 4, 7, 9, 13, 14, 18, 21, 24, 26, 31, 33, 38, 42, 46, 50, 54, 58, 61, 64, 70, 76, 81, 87, 92, 97, 99, 104, 106, 111, 118, 123, 127, 132, 136, 137, 144, 148, 155, 159, 163, 169, 173, 177, 181, 184, 190, 193, 199, 205, 211, 218, 226, 232, 238, 241, 247, 253

Offset: 1

Labos Elemer, Feb 28 2000

nonn

Analogous to A053048.

			For n = 6, A002110(6) = 30030, the corresponding iteration chain is {30030, 5760, 1536, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1}. The first power of 2 is the 4th item after 3 iterations. It is 512, therefore a(6) = log_2(512) = 9 and a(6) + 1 = 10 iterations is needed to reach the stationary value = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A002110, A053048, A053098.

a[n_] := Max@ IntegerExponent[ FixedPointList[ EulerPhi, Times @@ Prime[ Range@ n]], 2]; Array[a, 60] (* Giovanni Resta, May 30 2018 *)

a(n) = {my(p = prod(i=1, n, prime(i))); while(p >> valuation(p, 2) > 1, p = eulerphi(p)); valuation(p, 2);} \\ Amiram Eldar, Nov 19 2024

a(n) = log_2(A053098(n)). - Amiram Eldar, Nov 19 2024

cp's OEIS Frontend

A053047 a(n) is the first (and maximal) power of 2 arising during iterations of the Euler phi function with initial value n!.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A053099 When the Euler phi function is iterated with initial value A002110(n) = n-th primorial, a(n) = exponent of largest power of 2 arising in the iteration.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula