Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #24 Dec 31 2020 11:11:15
%S 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,
%T 40,40,42,44,46,46,50,52,54,56,58,60,64,64,66,68,70,72,76,78,80,80,82,
%U 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
%N Total number of odd divisors of all positive integers <= n, minus the total number of middle divisors of all positive integers <= n.
%C Conjecture 1: a(n) is also twice the total number of partitions of all positive integers <= n into an even number of consecutive parts.
%C 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.
%C For more information about the "subparts" of the symmetric representation of sigma see A279387 and A237593.
%F Conjecture: a(n) = A060831(n) - A240542(n).
%F Conjecture: a(n) = 2*A285902(n).
%t Accumulate@ Table[DivisorSum[n, 1 &, OddQ] - DivisorSum[n, 1 &, Sqrt[n/2] <= # < Sqrt[2 n] &], {n, 78}] (* _Michael De Vlieger_, May 18 2017 *)
%Y Partial sums of A281009.
%Y Cf. A060831, A196020, A235791, A236104, A237048, A237591, A237593, A240542, A245092, A279387, A285901, A285902.
%K nonn
%O 1,3
%A _Omar E. Pol_, May 14 2017