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

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127

Offset: 1

Author

Amiram Eldar, Oct 28 2019

Keywords

• nonn

Comments

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.

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.

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, ....

In particular, all the terms are odd since 2 is not a term (phi(1) = phi(2)).

The number of terms below 10^k for k = 1, 2, ... are 5, 49, 488, 4860, 48598, 485807, 4857394, 48572251, 485716764, 4857144075, ...

Apparently the sequence has an asymptotic density of 0.4857...

Examples

			3 is a term since it has 2 divisors, 1 and 3, and phi(1) = 1 != phi(3) = 2.
15 is a term since the phi values of its divisors, {1, 3, 5, 15}, are distinct: {1, 2, 4, 8}.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A000010, A032741, A102190, A319695, A319696.

Programs

Maple

filter:= proc(n) local D;
  D:=numtheory:-divisors(n);
  nops(D) = nops(map(numtheory:-phi,D))
end proc:
select(filter, [seq(i,i=1..200,2)]); # Robert Israel, Oct 29 2019

Mathematica

aQ[n_] := Length @ Union[EulerPhi /@ (d = Divisors[n])] == Length[d];  Select[Range[130], aQ]

PARI

isok(k) = #Set(apply(x->eulerphi(x), divisors(k))) == numdiv(k); \\ Michel Marcus, Oct 28 2019

Formula

Numbers k such that A319696(k) = A000005(k).

Numbers k such that A319695(k) = A032741(k).

Numbers k such that the k-th row of A102190 has distinct terms.