Say $n=p_1^{e_1}p_2^{e_2}\ldots p_k^{e_k}$ is such an integer. Then the sum of its divisors is, by the usual formula, $(1+p_1+p_1^2+\ldots+p_1^{e_1})(1+p_2+p_2^2+\ldots+p_2^{e_2})\ldots(1+p_k+\ldots+p_k^{e_k})$. But this is odd, so each term in the product is odd. But since no $p_i$ is equal to 2, we have $1+p_i+p_i^2+\ldots+p_i^{e_i}\equiv1+1+\ldots+1=e_i+1\mod 2$. Thus $e_i$ must be even for all $i$, and so $n$ must b a perfect square. Conversely, the above calculation shows that any odd perfect square has sum of divisors odd, so the $n$ that work are $1, 9, 25, 49, 81, 121, 169, 225, 289, 361, 441, 529, 625, 729, 841, 961, 1089, 1225, 1369, 1521, 1681,$ and $1849$.