With the prevalence of orthogonal frequency-division multiplexing (OFDM) in many standards, e.g., IEEE 802.11, IEEE 802.16, DVB-T, and DVB-T2, a number of variant modulation schemes based on OFDM have been proposed, which resort to signal sparsity to further enhance spectral efficiency and mitigate the high peak-to-average ratio (PAPR) problem. Among these variants, OFDM with subcarrier number modulation (OFDM-SNM) has been proven to be efficient for simple communication systems with low constellation modulation orders and limited decoding capability. To rigorously verify the performance advantages of OFDM-SNM, we present the study of OFDM-SNM in this paper from the information-theoretic perspective. In particular, we determine an upper bound on the mutual information of OFDM-SNM in closed form by using the log sum inequality. Also, we analyze the optimal pattern utilization probabilities (PUPs) for OFDM-SNM by channel-dependent coding and propose an easy-to-implement iterative algorithm to approach the optimal PUPs. Moreover, considering the practical achievability, we propose a Huffman coding based achievable PUP vector construction scheme to obtain the achievable PUPs and the corresponding achievable rate. We carry out numerical simulations to verify the effectiveness of this study and illustrate the efficiency of the obtained PUPs in comparison with several benchmarks.