|
|
A303435
|
|
Numbers n such that uphi(n) (the unitary totient function A047994) is a power of the number of unitary divisors of n (A034444).
|
|
0
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
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.
|
|
LINKS
|
|
|
EXAMPLE
|
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.
|
|
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
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|