This paper studies the problem of primality testing for numbers of the form h · 2n ± 1, where h < 2n is odd, and n is a positive integer. The authors describe a Lucasian primality test for these numbers in certain cases, which runs in deterministic quasi-quadratic time. In particular, the authors construct a Lucasian primality test for numbers of the form 3 · 5 · 17 · 2n ± 1, where n is a positive integer, in half of the cases among the congruences of n modulo 12, by means of a Lucasian sequence with a suitable seed not depending on n. The methods of Bosma (1993), Berrizbeitia and Berry (2004), Deng and Huang (2016) can not test the primality of these numbers.