A303435 Numbers n such that uphi(n) (the unitary totient function A047994) is a power of the number of unitary divisors of n (A034444).

1, 2, 3, 5, 9, 10, 17, 30, 34, 85, 170, 257, 514, 765, 1285, 1542, 4369, 8738, 39321, 65537, 131070, 131074, 327685, 655370, 1114129, 2949165, 3342387, 16843009, 33686018, 100271610, 151587081, 572662306, 2863311530

Offset: 1

Author

Amiram Eldar, Apr 24 2018

Keywords

• nonn

Comments

The unitary version of A289276.

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.

Examples

			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.
		

Crossrefs

Cf. A019434, A034444, A047994, A092506, A289276.

Programs

Mathematica

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