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%I #49 Dec 14 2021 10:06:07
%S 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,
%T 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,
%U 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
%N Characteristic function of numbers that have middle divisors.
%C Also characteristic function of numbers k whose symmetric representation of sigma(k) has an odd number of parts.
%C In other words: characteristic function of numbers k whose symmetric representation of sigma(k) has two vertices on its axis of symmetry.
%C a(n) is also the parity of the number of parts in the symmetric representation of sigma(n).
%H Antti Karttunen, <a href="/A347950/b347950.txt">Table of n, a(n) for n = 1..65537</a>
%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>
%F a(n) = A000035(A237271(n)).
%F From _Omar E. Pol_, Oct 30 2021: (Start)
%F a(n) = A348364(n) - 1.
%F a(n) = 1 - A348327(n). (End)
%e For n = 14 the divisors of 14 are [1, 2, 7, 14]. There are no middle divisors of 14, so a(14) = 0.
%e 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.
%e 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.
%e 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.
%t a[n_] := Boole[DivisorSum[n, 1 &, n/2 <= #^2 < 2*n &] > 0]; Array[a, 100] (* _Amiram Eldar_, Oct 01 2021 *)
%o (PARI) a(n) = sumdiv(n, d, my(d2 = d^2); (n/2 < d2) && (d2 <= n<<1)) > 0; \\ _Michel Marcus_, Oct 05 2021
%o (Python)
%o from sympy import divisors
%o def a(n): return 1*any(n/2<=d*d<2*n for d in divisors(n, generator=True))
%o print([a(n) for n in range(1, 106)]) # _Michael S. Branicky_, Oct 12 2021
%Y Characteristic function of A071562.
%Y Parity of A237271.
%Y Partial sums give A348110.
%Y Cf. A000035, A067742, A071090, A237048, A237270, A237591, A237593, A240542, A281007, A299761, A303297, A348327, A348364.
%K nonn
%O 1
%A _Omar E. Pol_, Sep 30 2021
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