A Proth number is a number of the form N = k·2^n + 1 for odd k, n a positive integer, and 2^n>k. The 2^n>k condition is needed since otherwise, every odd number >1 would be a Proth number. The first few Proth numbers are 3, 5, 9, 13, 17, 25, 33, 41, 49, 57, 65, ... (OEIS A080075). The Cullen numbers are a special case of the Proth numbers with k = n (and the inequality restriction dropped). The Fermat numbers are a special case of the Proth numbers with k = 1.

Cullen number | Cunningham number | Proth prime | Sierpiński number of the second kind | Sierpiński's composite number theorem