Powers as sums of consecutive odd numbers, and the odd numbers they skip

For odd k, n^k is the sum of n^((k-1)/2) consecutive odd numbers. Listing these blocks for n = 1, 2, 3, ..., we study the number of odd numbers skipped between consecutive blocks. For k = 5 the skips are the triangular numbers; for k = 7 they are three times the square pyramidal numbers; for k = 11 they are fifteen times a weighted sum of octahedral numbers (OEIS A085461). In each of these cases every skip is a multiple of the first skip 2^((k-3)/2) - 1. We prove that k = 5, 7, 11 are the only odd exponents with this property, using Fermat's little theorem and Zsigmondy's theorem.