Oscillometric finger pressing is a potential method for absolute blood pressure (BP) monitoring via a smartphone. The user presses their fingertip against a photoplethysmography-force sensor unit on a smartphone to steadily increase the external pressure on the underlying artery. Meanwhile, the phone guides the finger pressing and computes systolic BP (SP) and diastolic BP (DP) from the measured blood volume oscillations and finger pressure. The objective was to develop and evaluate reliable finger oscillometric BP computation algorithms. The collapsibility of thin finger arteries was exploited in an oscillometric model to develop simple algorithms for computing BP from the finger pressing measurements. These algorithms extract features from "width" oscillograms (oscillation width versus finger pressure functions) and the conventional "height" oscillogram for markers of DP and SP. Finger pressing measurements were obtained using a custom system along with reference arm cuff BP measurements from 22 subjects. Measurements were also obtained during BP interventions in some subjects for 34 total measurements. An algorithm employing the average of width and height oscillogram features predicted DP with correlation of 0.86 and precision error of 8.6mmHg with respect to the reference measurements. Analysis of arm oscillometric cuff pressure waveforms from an existing patient database provided evidence that the width oscillogram features are better suited to finger oscillometry. Analysis of oscillation width variations during finger pressing can improve DP computation. The study findings may help in converting widely available devices into truly cuffless BP monitors for improving hypertension awareness and control.