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A053044
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a(n) is the number of iterations of the Euler totient function to reach 1, starting at n!.
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7
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0, 1, 2, 4, 6, 8, 10, 13, 15, 18, 21, 24, 27, 30, 33, 37, 41, 44, 47, 51, 54, 58, 62, 66, 70, 74, 77, 81, 85, 89, 93, 98, 102, 107, 111, 115, 119, 123, 127, 132, 137, 141, 145, 150, 154, 159, 164, 169, 173, 178, 183, 188, 193, 197, 202, 207, 211, 216, 221, 226, 231
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OFFSET
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1,3
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COMMENTS
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Powers of 2 arise at the end of iteration chains without interruption. Analogous to A053025 and A053034. The order of speed of convergence is as follows: A000005 > A000010 > A051953: e.g., for 20! the lengths of the corresponding iteration chains are 6, 51, and 101, respectively.
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LINKS
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FORMULA
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EXAMPLE
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For n=1, no iteration is needed, so a(1)=0;
for n=2, the initial value is 2! = 2, so phi() must be applied once, thus a(2)=1;
for n=8, the iteration chain is {40320, 9216, 3072, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1}; its length = 14 = a(8) + 1, so the number of iterations applied to reach 1 is a(8)=13.
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MATHEMATICA
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Table[Length@ NestWhileList[EulerPhi, n!, # > 1 &] - 1, {n, 61}] (* or *)
Table[Length@ FixedPointList[EulerPhi, n!] - 2, {n, 61}] (* Michael De Vlieger, Jan 01 2017 *)
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PROG
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(PARI) a(n) = {my(nb = 0, ns = n!); while (ns != 1, ns = eulerphi(ns); nb++); nb; } \\ Michel Marcus, Jan 01 2017
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CROSSREFS
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KEYWORD
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nonn,changed
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AUTHOR
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STATUS
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approved
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