A348336 - OEIS

%I #19 Oct 29 2021 09:09:41

%S 0,0,1,1,2,2,3,3,3,4,5,5,6,7,7,7,8,8,9,9,10,11,12,12,12,13,14,14,15,

%T 15,16,16,17,18,18,18,19,20,21,21,22,22,23,24,24,25,26,26,26,26,27,28,

%U 29,29,30,30,31,32,33,33,34,35,35,35,36,36,37,38,39,39,40,40,41,42,43,44

%N Number of positive integers <= n that have no middle divisors.

%C a(n) is also the number of positive integers k <= n whose symmetric representation of sigma(k) has an even number of parts.

%e For n = 10 there are four positive integers <= 10 that have no middle divisors, they are [3, 5, 7, 10], so a(10) = 4.

%e On the other hand for n = 10 there are four positive integers k <= 10 whose symmetric representation of sigma(k) has an even number of parts, they are [3, 5, 7, 10], so a(10) = 4.

%t f[n_] := Boole[DivisorSum[n, 1 &, n/2 <= #^2 < 2*n &] == 0]; Accumulate @ Array[f, 100] (* _Amiram Eldar_, Oct 13 2021 *)

%o (PARI) is(n, f=factor(n))=my(t=(n+1)\2); fordiv(f, d, if(d^2>=t, return(d^2>2*n))); 0 ; \\ A071561

%o a(n) = sum(k=1, n, is(k)); \\ _Michel Marcus_, Oct 14 2021

%Y Partial sums of A348327.

%Y Cf. A067742, A071090, A071540, A071561, A071562, A071563, A237048, A237270, A237271, A237591, A237593, A240542, A281007, A299761, A303297, A340833, A346868, A347950, A348110.

%K nonn

%O 1,5

%A _Omar E. Pol_, Oct 13 2021