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Conjecture: Except for 1 and 3, all members of the sequence are even. If n is odd, it cannot be square-freesquarefree.

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Vincenzo Librandi, <a href="/A219930/b219930.txt">Table of n, a(n) for n = 1..200</a>

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Conjecture : If n is in the sequence, then the sequence contains an infinite number of multiples of n.

Conjecture : Except for 1 and 3, all members of the sequence are even. If n is odd, it cannot be square-free.

Conjecture : There does not exist N such that for all n > N, a(n) is divisible by 30.

nn = 78!; t = Table[EulerPhi[n], {n, nn}]; min = Infinity; t2 = {}; Do[If[t[[n]] <= min, AppendTo[t2, {n, t[[n]]}]; min = t[[n]]], {n, Length[t], 1, -1}]; t2 = Reverse[t2]; t3 = {}; mx = 0; Do[If[i[[2]] > mx, mx = i[[2]]; AppendTo[t3, i[[1]]]], {i, t2}]; t3 (* T. D. Noe, Dec 04 2012 *)

1, 3, 8, 14, 20, 36, 48, 66, 70, 96, 126, 132, 156, 240, 252, 300, 336, 450, 480, 540, 660, 690, 714, 870, 900, 1080, 1320, 1470, 1530, 1710, 1950, 2340, 2940, 2970, 3360, 3780, 4200, 4830, 5040, 5610, 5670, 5880, 6270, 7140, 7350, 7410, 8400, 9660, 9870

nn = 7!; t = Table[EulerPhi[n], {n, nn}]; min = Infinity; t2 = {}; Do[If[t[[n]] <= min, AppendTo[t2, {n, t[[n]]}]; min = t[[n]]], {n, Length[t], 1, -1}]; t2 = Reverse[t2]; t3 = {}; mx = 0; Do[If[i[[2]] > mx, mx = i[[2]]; AppendTo[t3, i[[1]]]], {i, t2}]; t3 (* T. D. Noe, Dec 04 2012 *)

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phi(1) = 1, and for n>=1, phi(n) never drops below >=1 again.

phi(3) = 2, and for n>=3, phi(n) never drops below >=2 again.

phi(8) = 4, and for n>=8, phi(n) never drops below >=4 again.

phi(14) = 6, and for n>=14, phi(n) never drops below >=6 again.

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