A348327 Characteristic function of numbers that have no middle divisors.

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

Offset

1

Comments

Also characteristic function of numbers k whose symmetric representation of sigma(k) has an even number of parts.

In other words: characteristic function of numbers k whose symmetric representation of sigma(k) has only one vertex on its axis of symmetry.

a(n) is also the parity of the number of vertices 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(A340833(n)).

a(n) = A000035(A348364(n)).

a(n) = 1 - A347950(n).

a(n) = [A067742(n) == 0], where [ ] is the Iverson bracket. - Antti Karttunen, Oct 14 2021

a(n) = 2 - A348364. - Omar E. Pol, Oct 30 2021

Example

For n = 14 the divisors of 14 are [1, 2, 7, 14]. There are no middle divisors of 14, so a(14) = 1.
On the other hand the symmetric representation of sigma(14) has two parts [12, 12]. The number of parts is even, so a(14) = 1.
For n = 15 the divisors of 15 are [1, 3, 5, 15]. There are two middle divisors of 15: [3, 5], so a(15) = 0.
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) = 0.

Mathematica

a[n_] := Boole[DivisorSum[n, 1 &, n/2 <= #^2 < 2*n &] == 0]; Array[a, 100] (* Amiram Eldar, Oct 13 2021 *)

Prog

(PARI)
A347950(n) = ((sumdiv(n, d, my(d2 = d^2); (n/2 < d2) && (d2 <= n<<1))) > 0); \\ From A347950
A348327(n) = (1-A347950(n)); \\ Antti Karttunen, Dec 13 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 14 2021

Crossrefs

Characteristic function of A071561.

Parity of A340833 and of A348364.

Partial sums give A348336.

Cf. A000035, A067742, A071090, A071540, A071562, A071563, A237048, A237270, A237271, A237591, A237593, A240542, A281007, A299761, A303297, A346868, A347950.

Sequence in context: A358770 A022925 A327211 * A144607 A327253 A051840

Adjacent sequences: A348324 A348325 A348326 * A348328 A348329 A348330

Keyword

nonn

Author

Omar E. Pol, Oct 12 2021

Status

approved