A285999 Total number of odd divisors of all positive integers <= n, minus the total number of middle divisors of all positive integers <= n.

0, 0, 2, 2, 4, 4, 6, 6, 8, 10, 12, 12, 14, 16, 18, 18, 20, 22, 24, 24, 28, 30, 32, 32, 34, 36, 40, 40, 42, 44, 46, 46, 50, 52, 54, 56, 58, 60, 64, 64, 66, 68, 70, 72, 76, 78, 80, 80, 82, 84, 88, 90, 92, 94, 98, 98, 102, 104, 106, 108, 110, 112, 116, 116, 120, 122, 124, 126, 130, 132, 134, 134, 136, 138, 144, 146, 148, 152

Offset: 1

Author

Omar E. Pol, May 14 2017

Keywords

• nonn

Comments

Conjecture 1: a(n) is also twice the total number of partitions of all positive integers <= n into an even number of consecutive parts.

Conjecture 2: a(n) is also the total number of equidistant subparts of the symmetric representations of sigma of all positive integers <= n. Thus a(n) is also the total number of equidistant subparts in the terraces of the stepped pyramid with n levels described in A245092.

For more information about the "subparts" of the symmetric representation of sigma see A279387 and A237593.

Crossrefs

Partial sums of A281009.

Cf. A060831, A196020, A235791, A236104, A237048, A237591, A237593, A240542, A245092, A279387, A285901, A285902.

Programs

Mathematica

Accumulate@ Table[DivisorSum[n, 1 &, OddQ] - DivisorSum[n, 1 &, Sqrt[n/2] <= # < Sqrt[2 n] &], {n, 78}] (* Michael De Vlieger, May 18 2017 *)

Formula

Conjecture: a(n) = A060831(n) - A240542(n).

Conjecture: a(n) = 2*A285902(n).