A348336 Number of positive integers <= n that have no middle divisors.

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, 15, 16, 16, 17, 18, 18, 18, 19, 20, 21, 21, 22, 22, 23, 24, 24, 25, 26, 26, 26, 26, 27, 28, 29, 29, 30, 30, 31, 32, 33, 33, 34, 35, 35, 35, 36, 36, 37, 38, 39, 39, 40, 40, 41, 42, 43, 44

Offset

1,5

Comments

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

Links

Table of n, a(n) for n=1..76.

Example

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

Mathematica

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

Prog

(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
a(n) = sum(k=1, n, is(k)); \\ Michel Marcus, Oct 14 2021

Crossrefs

Partial sums of A348327.

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

Sequence in context: A125059 A369611 A029112 * A302984 A029094 A262950

Adjacent sequences: A348333 A348334 A348335 * A348337 A348338 A348339

Keyword

nonn

Author

Omar E. Pol, Oct 13 2021

Status

approved