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A326835 Numbers whose divisors have distinct values of the Euler totient function (A000010).

%I A326835 #36 Oct 29 2019 10:41:54
%S A326835 1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,
%T A326835 49,51,53,55,57,59,61,65,67,69,71,73,75,77,79,81,83,85,87,89,91,93,95,
%U A326835 97,99,101,103,105,107,109,111,113,115,117,119,121,123,125,127
%N A326835 Numbers whose divisors have distinct values of the Euler totient function (A000010).
%C A326835 Since Sum_{d|k} phi(d) = k, these are numbers k such that the set {phi(d) | d|k} is a partition of k into distinct parts.
%C A326835 Includes all the odd prime numbers, since an odd prime p has 2 divisors, 1 and p, whose phi values are 1 and p-1.
%C A326835 If k is a term, then all the divisors of k are also terms. If k is not a term, then all its multiples are not terms. The primitive terms of the complementary sequence are 2, 63, 273, 513, 585, 825, 2107, 2109, 2255, 3069, ....
%C A326835 In particular, all the terms are odd since 2 is not a term (phi(1) = phi(2)).
%C A326835 The number of terms below 10^k for k = 1, 2, ... are 5, 49, 488, 4860, 48598, 485807, 4857394, 48572251, 485716764, 4857144075, ...
%C A326835 Apparently the sequence has an asymptotic density of 0.4857...
%H A326835 Robert Israel, <a href="/A326835/b326835.txt">Table of n, a(n) for n = 1..10000</a>
%F A326835 Numbers k such that A319696(k) = A000005(k).
%F A326835 Numbers k such that A319695(k) = A032741(k).
%F A326835 Numbers k such that the k-th row of A102190 has distinct terms.
%e A326835 3 is a term since it has 2 divisors, 1 and 3, and phi(1) = 1 != phi(3) = 2.
%e A326835 15 is a term since the phi values of its divisors, {1, 3, 5, 15}, are distinct: {1, 2, 4, 8}.
%p A326835 filter:= proc(n) local D;
%p A326835   D:=numtheory:-divisors(n);
%p A326835   nops(D) = nops(map(numtheory:-phi,D))
%p A326835 end proc:
%p A326835 select(filter, [seq(i,i=1..200,2)]); # _Robert Israel_, Oct 29 2019
%t A326835 aQ[n_] := Length @ Union[EulerPhi /@ (d = Divisors[n])] == Length[d];  Select[Range[130], aQ]
%o A326835 (PARI) isok(k) = #Set(apply(x->eulerphi(x), divisors(k))) == numdiv(k); \\ _Michel Marcus_, Oct 28 2019
%Y A326835 Cf. A000005, A000010, A032741, A102190, A319695, A319696.
%K A326835 nonn
%O A326835 1,2
%A A326835 _Amiram Eldar_, Oct 28 2019