A347950 Characteristic function of numbers that have middle divisors.

1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0

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

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for characteristic functions

Formula

a(n) = A000035(A237271(n)).

From Omar E. Pol, Oct 30 2021: (Start)

a(n) = A348364(n) - 1.

a(n) = 1 - A348327(n). (End)

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

Characteristic function of A071562.

Parity of A237271.

Partial sums give A348110.

Cf. A000035, A067742, A071090, A237048, A237270, A237591, A237593, A240542, A281007, A299761, A303297, A348327, A348364.

Sequence in context: A189292 A331168 A331281 * A105470 A359824 A087429

Adjacent sequences: A347947 A347948 A347949 * A347951 A347952 A347953

Keyword

nonn

Author

Omar E. Pol, Sep 30 2021

Status

approved