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.
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
Keywords
Examples
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.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
Programs
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Mathematica
a[n_] := Max@ IntegerExponent[ FixedPointList[ EulerPhi, Times @@ Prime[ Range@ n]], 2]; Array[a, 60] (* Giovanni Resta, May 30 2018 *)
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PARI
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
Formula
a(n) = log_2(A053098(n)). - Amiram Eldar, Nov 19 2024
Comments