A053047 -id:A053047 - cp's OEIS frontend

A053048 a(n) is the number of terminal iterations applied to powers of 2 arising in the iterations of the Euler phi function with initial value n!.

0, 1, 1, 3, 5, 6, 7, 10, 10, 13, 16, 18, 20, 22, 24, 28, 32, 33, 34, 38, 39, 43, 47, 50, 54, 57, 57, 60, 63, 66, 69, 74, 77, 82, 85, 87, 89, 91, 93, 98, 103, 105, 107, 112, 114, 119, 124, 128, 130, 135, 139, 143, 147, 148, 153, 157, 158, 162, 166, 170, 174, 178, 179

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 8192, after which phi is applied 13 additional times to reach the stationary value 1.

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

Cf. A000010, A000142, A048855, A053039, A053047.

a[n_] := Max@ IntegerExponent[ FixedPointList[ EulerPhi, n!], 2]; Array[a, 63] (* Giovanni Resta, May 30 2018 *)

a(n) = log_2(A053047(n)). - Amiram Eldar, Aug 17 2024

A053098 When the Euler phi function is iterated with initial value primorial(n) = A002110(n), a(n)= first (maximum) power of 2 emerging during iteration.

Original entry on oeis.org

2, 2, 8, 16, 128, 512, 8192, 16384, 262144, 2097152, 16777216, 67108864, 2147483648, 8589934592, 274877906944, 4398046511104, 70368744177664, 1125899906842624, 18014398509481984, 288230376151711744, 2305843009213693952, 18446744073709551616, 1180591620717411303424

Offset: 1

Labos Elemer, Feb 28 2000

nonn

Analogous to A053047 (the same with a similar large initial value, n!).

			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) = 512.

Amiram Eldar, Table of n, a(n) for n = 1..500

Cf. A000010, A002110, A053047, A053099.

a[n_] := 2^ Max@ IntegerExponent[ FixedPointList[ EulerPhi, Times @@ Prime[Range[n]]], 2]; Array[a, 20] (* Giovanni Resta, May 30 2018 *)

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

a(n) = 2^A053099(n). - Amiram Eldar, Nov 19 2024

a(21)-a(23) from Amiram Eldar, Nov 19 2024

cp's OEIS Frontend

A053048 a(n) is the number of terminal iterations applied to powers of 2 arising in the iterations of the Euler phi function with initial value n!.

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