AbstractFormulas are derived for n1/2‐point osculatory trigonometric interpolation, which give the unique trigonometric sum Sn(x) satisfying Sn(xi) = f(xi), i = 1,2, …, n, and S′n(xi) = f′(xi), i = 1,2, …, n + 1. In conjunction with previously derived formulas, we have predictor‐corrector formulas for stepwise numerical integration of y′ = Φ(x, y). These formulas might be applicable to numerical integration where y is either purely or nearly periodic (e. g., orbits, satellites, etc.). For each n, for xi equally spaced at intervals of h, there is a different predictor‐corrector pair for every h. To obtain predictor‐corrector formulas for y″ = Φ (x, y, y′), one requires n 1/3 ‐and n 2/3 ‐point hyperosculatory trigonometric interpolation formulas (i. e., based upon fi, f′i and f′i, i = 1,2, …, n, and one or two out of fn+1, f′n+1, and f″n+1). A procedure is outlined for obtaining unique n 1/3 ‐and n 2/3 ‐point formulas (different for odd and even n). Remainder formulas in terms of divided differences for complex arguments (not too informative) are given for some formulas having an odd number of interpolation conditions.