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%I #4 Jun 18 2018 18:09:30
%S 1,2,3,5,9,10,17,30,34,85,170,257,514,765,1285,1542,4369,8738,39321,
%T 65537,131070,131074,327685,655370,1114129,2949165,3342387,16843009,
%U 33686018,100271610,151587081,572662306,2863311530
%N Numbers n such that uphi(n) (the unitary totient function A047994) is a power of the number of unitary divisors of n (A034444).
%C The unitary version of A289276.
%C Since A034444(n)=2^omega(n) is a power of 2, all the terms are products of 2 and the Fermat primes (A019434), each with multiplicity < 2, except for 3 that may be of multiplicity of 2 (since 3^2 = 2^3 + 1). If there is no 6th Fermat prime, then this sequence is finite with 33 terms.
%e 2863311530 = 2 * 5 * 17 * 257 * 65537 is in the sequence since it has 2^5 unitary divisors, and its uphi value is 2^30 = (2^5)^6.
%t uphi[n_]:=If[n == 1,1,(Times @@ (Table[#[[1]]^#[[2]] - 1, {1}] & /@ FactorInteger [n]))[[1]]]; aQ[n_] := If[n == 1, True, IntegerQ[Log[2, uphi[n]]/PrimeNu[n]]]; v = Union[Times @@@ Rest[Subsets[{1, 2, 3, 5, 17, 257, 65537}]]]; w = Union[v, 3*v]; s = {}; Do[w1 = w[[k]]; If[aQ[w1], AppendTo[s, w1]], {k, 1, Length[w]}]; s
%Y Cf. A019434, A034444, A047994, A092506, A289276.
%K nonn
%O 1,2
%A _Amiram Eldar_, Apr 24 2018
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