A354452 Number of middle divisors of 2*n.

1, 1, 2, 1, 0, 2, 0, 1, 1, 2, 0, 2, 0, 2, 2, 1, 0, 1, 0, 2, 2, 0, 0, 2, 1, 0, 2, 2, 0, 2, 0, 1, 2, 0, 2, 3, 0, 0, 0, 2, 0, 2, 0, 2, 2, 0, 0, 2, 1, 1, 0, 2, 0, 2, 2, 2, 0, 0, 0, 4, 0, 0, 2, 1, 2, 2, 0, 0, 0, 2, 0, 3, 0, 0, 2, 0, 2, 2, 0, 2, 1, 0, 0, 2, 2, 0, 0, 2, 0, 4, 2, 0, 0, 0, 2, 2, 0, 1, 2, 1, 0, 2, 0, 2, 2

Offset: 1

Author

Omar E. Pol, May 30 2022

Keywords

• nonn

Comments

a(n) is the number of middle divisors of the n-th even number.

a(n) is also the width of the terrace at the level 2*n starting from the top in the main diagonal of the stepped pyramid described in A245092.

a(n) is also the number of central subparts in the symmetric representation of sigma(2n). For more information about the subparts see A279387.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..32768

Crossrefs

Bisection of A067742.

Cf. A005843, A099777, A237048, A237270, A237271, A237593, A245092, A249351, A279387, A319796, A354451.

Programs

Mathematica

a[n_] := DivisorSum[2*n, 1 &, n <= #^2 < 4*n &]; Array[a, 100] (* Amiram Eldar, Jun 01 2022 *)

PARI

A354452(n) = { n <<= 1; sumdiv(n, d, my(d2 = d^2); (n / 2 < d2 && d2 <= n << 1)); }; \\ Antti Karttunen, Jan 17 2025, after program in A067742 by M. F. Hasler, May 12 2008

Formula

a(n) = A067742(2n).

a(n) = A067742(A005843(n)).