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

1,3

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.

Links

Table of n, a(n) for n=1..78.

Formula

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

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

Mathematica

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

Crossrefs

Partial sums of A281009.

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

Sequence in context: A308767 A353504 A345419 * A005186 A259881 A238132

Adjacent sequences: A285996 A285997 A285998 * A286000 A286001 A286002

Keyword

nonn

Author

Omar E. Pol, May 14 2017

Status

approved