A348004 - OEIS

%I #18 Sep 25 2021 04:45:21

%S 1,3,4,5,7,8,9,11,12,13,15,16,17,19,20,21,23,24,25,27,28,29,31,32,33,

%T 35,36,37,39,40,41,43,44,45,47,48,49,51,52,53,55,56,57,59,60,61,63,64,

%U 65,67,68,69,71,72,73,75,76,77,79,80,81,83,85,87,88,89,91

%N Numbers whose unitary divisors have distinct values of the unitary totient function uphi (A047994).

%C First differs from A042965 \ {0} at n=63, and from A122906 at n=53.

%C Since Sum_{d|k, gcd(d,k/d)=1} uphi(d) = k, these are numbers k such that the set {uphi(d) | d|k, gcd(d,k/d)=1} is a partition of k into distinct parts.

%C Includes all the odd prime powers (A061345), since an odd prime power p^e has 2 unitary divisors, 1 and p^e, whose uphi values are 1 and p^e - 1. It also includes all the powers of 2, except for 2 (A151821).

%C If k is a term, then all the unitary divisors of k are also terms.

%C The number of terms not exceeding 10^k for k = 1, 2, ... are 7, 74, 741, 7386, 73798, 737570, 7374534, 73740561, 737389031, 7373830133, ... Apparently, this sequence has an asymptotic density 0.73738...

%H Amiram Eldar, <a href="/A348004/b348004.txt">Table of n, a(n) for n = 1..10000</a>

%F Numbers k such that A348001(k) = A034444(k).

%e 4 is a term since it has 2 unitary divisors, 1 and 4, and uphi(1) = 1 != uphi(4) = 3.

%e 12 is a term since the uphi values of its unitary divisors, {1, 3, 4, 12}, are distinct: {1, 2, 3, 6}.

%t f[p_, e_] := p^e - 1; uphi[1] = 1; uphi[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := Length @ Union[uphi /@ (d = Select[Divisors[n], CoprimeQ[#, n/#] &])] == Length[d]; Select[Range[100], q]

%o (Python)

%o from math import prod

%o from sympy.ntheory.factor_ import udivisors, factorint

%o A348004_list = []

%o for n in range(1,10**3):

%o     pset = set()

%o     for d in udivisors(n,generator=True):

%o         u = prod(p**e-1 for p, e in factorint(d).items())

%o         if u in pset:

%o             break

%o         pset.add(u)

%o     else:

%o         A348004_list.append(n) # _Chai Wah Wu_, Sep 24 2021

%Y The unitary version of A326835.

%Y Cf. A034444, A042965, A047994, A061345, A151821, A122906, A348001.

%K nonn

%O 1,2

%A _Amiram Eldar_, Sep 23 2021