A348004 Numbers whose unitary divisors have distinct values of the unitary totient function uphi (A047994).

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, 35, 36, 37, 39, 40, 41, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 56, 57, 59, 60, 61, 63, 64, 65, 67, 68, 69, 71, 72, 73, 75, 76, 77, 79, 80, 81, 83, 85, 87, 88, 89, 91

Offset: 1

Author

Amiram Eldar, Sep 23 2021

Keywords

• nonn

Comments

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

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.

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).

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

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...

Examples

			4 is a term since it has 2 unitary divisors, 1 and 4, and uphi(1) = 1 != uphi(4) = 3.
12 is a term since the uphi values of its unitary divisors, {1, 3, 4, 12}, are distinct: {1, 2, 3, 6}.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

The unitary version of A326835.

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

Programs

Mathematica

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]

Python

from math import prod
from sympy.ntheory.factor_ import udivisors, factorint
A348004_list = []
for n in range(1,10**3):
    pset = set()
    for d in udivisors(n,generator=True):
        u = prod(p**e-1 for p, e in factorint(d).items())
        if u in pset:
            break
        pset.add(u)
    else:
        A348004_list.append(n) # Chai Wah Wu, Sep 24 2021

Formula

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