OFFSET
1
COMMENTS
Also characteristic function of numbers k whose symmetric representation of sigma(k) has an odd number of parts.
In other words: characteristic function of numbers k whose symmetric representation of sigma(k) has two vertices on its axis of symmetry.
a(n) is also the parity of the number of parts in the symmetric representation of sigma(n).
LINKS
FORMULA
EXAMPLE
For n = 14 the divisors of 14 are [1, 2, 7, 14]. There are no middle divisors of 14, so a(14) = 0.
On the other hand the symmetric representation of sigma(14) has two parts: [12, 12]. The number of parts is even, so a(14) = 0.
For n = 15 the divisors of 15 are [1, 3, 5, 15]. There are two middle divisors of 15: [3, 5], so a(15) = 1.
On the other hand the symmetric representation of sigma(15) has three parts: [8, 8, 8]. The number of parts is odd, so a(15) = 1.
MATHEMATICA
a[n_] := Boole[DivisorSum[n, 1 &, n/2 <= #^2 < 2*n &] > 0]; Array[a, 100] (* Amiram Eldar, Oct 01 2021 *)
PROG
(PARI) a(n) = sumdiv(n, d, my(d2 = d^2); (n/2 < d2) && (d2 <= n<<1)) > 0; \\ Michel Marcus, Oct 05 2021
(Python)
from sympy import divisors
def a(n): return 1*any(n/2<=d*d<2*n for d in divisors(n, generator=True))
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Oct 12 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Omar E. Pol, Sep 30 2021
STATUS
approved