a(n) = number of m such that sum of proper divisors of m (A001065(m)) is 2*n+1.

1, 0, 1, 1, 1, 2, 2, 2, 2, 3, 2, 3, 2, 2, 5, 3, 3, 4, 1, 4, 5, 3, 3, 6, 4, 3, 6, 5, 3, 6, 3, 6, 6, 2, 5, 8, 4, 5, 7, 6, 4, 8, 5, 5, 9, 4, 4, 9, 3, 6, 8, 7, 5, 9, 6, 7, 10, 7, 5, 13, 5, 5, 11, 4, 8, 11, 5, 6, 8, 8, 7, 13, 5, 5, 12, 5, 8, 12, 4, 10, 10, 5, 5, 15

1

By separating A048138(n) into even and odd n, we obtain better information about how this sequence behaves. The even part is shown in S000868. See where records occur in S000872.

T. D. Noe, Plot of 10000 terms

T. D. Noe, Table of 10000 terms

Wikipedia, Divisor function

(Mma) nn = 200; s = Table[0, {nn}]; Do[k = DivisorSigma[1, n] - n; If[0 < k <= nn, s[[k]]++], {n, (nn/2)^2}]; Take[s, {3, nn, 2}]

Cf. A001065, A048138, S000868, S000872.

nonn

T. D. Noe, May 03 2016