OFFSET
1,2
COMMENTS
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...
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
EXAMPLE
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}.
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]
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Sep 23 2021
STATUS
approved